arxiv.org · arxiv:1306.2002v3 [math.ag] 9 feb 2015 apparent contours of nonsingular real cubic...

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APPARENT CONTOURS OF NONSINGULAR REAL CUBIC SURFACES

Sergey Finashin, Viatcheslav Kharlamov

We give a complete deformation classification of real Zariski sextics, that is of generic apparentcontours of nonsingular real cubic surfaces. As a by-product, we observe a certain ”reversion” dualityin the set of deformation classes of these sextics.

Contents

1. Introduction 31.1. The main problem and principal results 31.2. Partnership duality 41.3. Some related results 41.4. Contents of the paper 41.5. Terminology conventions and notation 51.6. Acknowledgements 5

2. Zariski sextics 52.1. Generic projection in the complex setting 52.2. Generic projection in the real setting 62.3. Restrictions on the arrangements of ovals and real cusps 72.4. The code of a Zariski sextic 92.5. The types I and II of (M-d)-curves 92.6. Reversion of Zariski sextics 102.7. The relation between the topology of cubic surfaces and their Zariski

sextics 112.8. Relation to the double covering cuspidal K3-surface Y 122.9. Deformation classification statement 13

3. Preliminaries on lattice theory 133.1. Even lattices and their discriminants 133.2. Groups with inner products and quadratic refinements 143.3. The p-components 153.4. The discriminant p-ranks of lattices 153.5. The Brown invariant 163.6. Elementary enhanced 2-groups 173.7. Elementary 3-groups with an inner product 183.8. Extensions of lattices 183.9. Gluing of lattices 193.10.The orthogonal complement of ±2-elements 203.11.Involutions via gluing 203.12.Stability 213.13.Different involutions with the same eigenlattices 22

2010 Mathematics Subject Classification. Primary 14P25, 14J28, 14J70, 14N25, 14H45.The second author acknowledges a financial support by the grant ANR-09-BLAN-0039-01 of Agence Nationale

de la Recherche.

1

http://arxiv.org/abs/1306.2002v3

2 S. FINASHIN, V. KHARLAMOV

4. Topology and arithmetics of the covering K3-surfaces 224.1. Covering K3 after desingularization 224.2. Deficiency in the Smith inequalities 234.3. Real varieties of type I 24

4.4. Resolution decoration of H2(Y ) 254.5. The Galois S3-coverings 264.6. Abstract K3-lattice conical (∆, h)-decorations 284.7. S0-eigenlattices 294.8. Geometric involutions 294.9. T-pairs and T-halves 304.10.T-halves and pairs of Mobius involutions 31

5. Arithmetics of geometric involutions 325.1. Automorphisms of 3-elementary inner product groups of small rank 325.2. Reduction homomorphism 335.3. Equivariant epistability of S0 335.4. Gluing of involutions 345.5. Realizability of T-pairs by geometric involutions 35

6. Deformation classes via periods 376.1. The complex period map 376.2. The real period map 406.3. Deformation classification via geometric involutions 416.4. From a trigonal curve in Σ4 to Zariski sextics in P 2 42

7. Arithmetics of the T-pairs 437.1. Geography of the ascending T-pairs 437.2. The list of T-halves 457.3. Stability of the T-halves 477.4. Property of (− 1

2 )-transitivity for the eigenlattices T− 508. Back to Zariski curves 51

8.1. Classification of geometric involutions 518.2. Reversion roots 528.3. Reversion partners of T-pairs 528.4. Classification of real Zariski sextics 538.5. The IDs of real Zariski sextics 548.6. Proof of Theorem 2.9.1 57

9. Concluding Remarks 579.1. Purely real statements 579.2. Transversal pairs of conic and cubic 589.3. Ordering of IDs 599.4. Nonsingular partners 599.5. Promiscuity 60

References 60

APPARENT CONTOURS OF NONSINGULAR REAL CUBIC SURFACES 3

“Tout n’est qu’apparence, non ?”

Alberto Giacometti

1. Introduction

1.1. The main problem and principal results. There exist many ways to visualize cubicsurfaces. One of those that take into account not only the internal geometry of the surface butalso its position in the space, consists in ”viewing the surface from an external point”, that is inconsidering central projections of the surface onto a plane and thus representing the surface asa 3-fold covering of the plane branched over a certain curve, called the apparent contour of thesurface. If the cubic surface is nonsingular, such an apparent contour is a curve of degree 6 whosesingular locus, for a generic choice of the center of projection, is formed by six cusps lying on aconic. Here, the condition of “generic choice” means certain transversality of the cubic surfaceto its first two polars with respect to the selected point, see Section 2.1 (in fact, the same kindof generic choice assumption is specific to the literature on Chisini conjecture, see, for example,[Mo] and [Ku1]).

The sextics with six cusps lying on a conic are called below Zariski sextics, following the“Arnold principle” [A]. The converse statement that any such a sextic is an apparent contourof some nonsingular cubic surface with respect to a generic center of projection is a classicalobservation (see [Sal], XIV.445 for the direct statement, and [Z1],[Z2],[Seg] for the converse one).This correspondence establishes an isomorphism between the space of projective classes of pairsformed by a nonsingular cubic surface with a generic center of projection, and the space ofprojective classes of Zariski sextics. Therefore, studying the former classes is reduced to studyingthe latter ones, and it works equally well over C and over R.

G. Mikhalkin had undertaken an analysis of the apparent contours of real nonsingular cubicsurfaces and reported the results he obtained in [M]. Namely, he looked for a topological clas-sification of the real apparent contours that are enhanced by specifying the topological type ofthe real locus of the cubic surface (such an enhancing can be expressed by coloring the part ofthe real plane where the projection is three-to-one). Mikhalkin listed 49 enhanced isotopy classesof apparent contours that he constructed, mentioned 7 enhanced isotopy classes whose existenceis uncertain, and claimed that there are no others. Our research resulted from attempts to un-derstand Mikhalkin’s results, to find the proofs, to complete the classification and, overall, tosharpen it by providing a classification up to equisingular deformations. Recall that for planereduced curves, which is our case, equisingular deformations can be defined in topological terms,as continuous families of algebraic curves preserving the quantity of singular points and theirMilnor numbers. Thus, in our setting the equisingular deformation classes of the apparent con-tours of nonsingular real cubic surfaces are nothing but the connected components of the spaceof real Zariski sextics.

To obtain such a deformation classification of apparent contours, we analyze the K3-surfacesthat we obtain as double covers of the plane branched along Zariski curves. Kulikov’s theoremon surjectivity of the period map for K3-surfaces (see [Ku2]) allows us to reduce the deformationclassification to a classification of certain involutions on the K3-lattice (so called ”geometric invo-lutions”, see Section 4.8 for precise definitions). Then, using Nikulin’s results on the arithmeticsof integral lattices (see [N1]), and the extension of these results by Miranda-Morrison ([MM1],[MM2]) we make the final classification explicit.

As a result, we prove that there exist precisely 68 deformation classes of apparent contours.In turn, we find that 7 Mikhalkin’s uncertain enhanced isotopy classes are actually realizableand that, moreover, there exist 6 more enhanced isotopy classes missing in his list; the full list ofenhanced isotopy classes contains 62 items. The difference, also equal to 6, between the numberof deformation classes and the number of enhanced isotopy classes is due to existence of 6 pairs-twins of deformation classes, such that the twins in each pair give the same isotopy class of


apparent contours and the same topological type of the cubic surface, but differ by the complextype of the underlying Zariski sextic, which can be dividing or not dividing.

The final classification is presented at two levels. At the first level, we establish a one-to-onecorrespondence between the set of deformation classes and the set of conjugacy classes of whatwe call ascending geometric involutions on the K3-lattice (see Theorem 6.3.1) and enumeratethe latter conjugacy classes in terms of the eigenlattices of involutions (see Theorem 7.3.11).Such a classification can be viewed as a kind of ”imaginary” one, since it does not immediatelydisclose the topology of the corresponding apparent contours. At the second level, we translatethe information on lattices into a “real” information, which yields a classification in terms ofthe ID’s of the apparent contours (see Theorem 2.9.1), where an ID is, roughly speaking, justan enhanced isotopy type with an additional bit of information specifying whether a sextic isdividing or not.

1.2. Partnership duality. The list of the ID’s of Zariski sextics that we give in Theorem 2.9.1is organized so that it emphasizes some duality resulted from the classification. This dualitysplits 62 of the deformation classes into 31 pairs (so that only 6 remaining deformation classesare left without a pair). Geometrically, in terms of arrangement of components (and their cusps)of a real Zariski sextic, this duality can be described as a certain reversion, see details in Section2.6. In Section 6.4 we give a “conceptual” explanation of this duality via a “wall crossing” ofspecial faces that can be found on all but six exceptional fundamental period domains in theperiod space of our K3-surfaces. Such special faces correspond to degeneration of Zariski sexticsto a triple conic by means of families of the form Q3 + tf2Q

2 + t2f4Q + t3f6 (more precisely,of the form Q3 + t(f1(x, y, z)Q+ tf3(x, y, z))

2 = 0), and the K3-surface resulting in the limit ofsuch a family at t = 0 is an appropriate double cover of the ruled surface Σ4, so that the dualityin terms of these K3 surfaces results in twisting the real structure by the deck transformation ofthe covering, or equivalently, in terms of the families, it results in changing the sign of t, that isin switching to Q3 − tf2Q

2 + t2f4Q− t3f6 (which is equivalent to switching the sign of Q).

1.3. Some related results. The approach to studying the topology and deformation classesof real K3-surfaces and, in particular, of nonsingular real plane sextics via the period map wasdeveloped by V. Kharlamov [Kh2] and V. Nikulin [N1] in the 70th. Later on, in the 90th, thesame approach was used by I. Itenberg [It] for topological and deformation study of real planesextics with one node.

Applications of a similar approach to complex K3-surfaces are much more abundant: here, wemention only the works by T. Urabe [U] on classification of configurations of simple singulari-ties on complex plane sextics and the recent works by A. Degtyarev [D1]-[D2] on deformationclassification of complex plane sextics with simple singularities.

The Zariski sextics, which are the subject of our paper, belong to the so called class of planecurves of torus type: they are generic plane curves of torus type of degree (2, 3). There exists avast literature on the geometry of plane curves of torus type over the complex field. In particular,due to works by D.T. Pho and M. Oka a complete classification of configurations of singularitieson complex sextics of torus type is actually known, see [OP] and [O].

1.4. Contents of the paper. The paper is organized as follows. In Chapter 2 we introduceZariski sextics, analyze their relation to cubic surfaces, and study the basic properties of Zariskisextics and of the double planes branched along them. The chapter is concluded with our principaldeformation classification Theorem 2.9.1. In Chapter 3, which is devoted to the arithmetics ofintegral lattices, we recall the basic definitions and some well-known results on lattices (theirdiscriminant finite forms, gluing of lattices and involutions), slightly developing them and makinga special emphasis on the lattices that have only discriminant factors 2 and 3, since it is this kindof lattices that appears later on in the proofs of the main results. Chapter 4 deals with our mainobject of investigation, the K3-surfaces that are double coverings of the plane branched along


Zariski sextics. Here, we relate the topological and geometrical properties of these surfaces withthe arithmetics of K3-lattices. In particular, we introduce and study geometric involutions onthe K3-lattices and associated with them certain T -pairs of eigenlattices.

Chapter 5 starts with a study of the action of geometric involutions on the lattice generatedby the exceptional divisors and ends with proving the realizability of all the T -pairs via geometricinvolutions. In Chapter 6 we introduce the period space for K3-surfaces that are double coveringsof the plane branched along Zariski sextics, check that up to codimension two the periods of ourK3-surfaces fill out the fundamental domains of a group generated by reflections, and deducefrom that the bijection between the set of deformation classes of real Zariski sextics and the setof conjugacy classes of ascending geometric involutions. In Chapter 7 we classify the ascendingT -pairs and prove their stability. Finally, in Chapter 8 we apply the results of preceding Chaptersand enumerate the geometric involutions, and hence the deformation classes of real Zariski sextics,in terms of T -pairs. We conclude with translating the classification into the language of IDs ofreal Zariski sextics, which proves the deformation classification statement formulated in Chapter2. A few final remarks are collected in Chapter 9.

1.5. Terminology conventions and notation. A real algebraic variety is always consideredas a pair (X, c), where X is a complex one and c : X → X is an anti-holomorphic involution calledthe complex conjugation, or the real structure. The locus of complex points of X is denoted byX(C), and the real locus, Fix c, by X(R). In some cases, when it causes no confusion, we denotealso by c the induced involution in the homology.

When we speak on singular points and on (equisingular) deformations, we take into accountnot only the real points but the complex ones as well. For example, from such a viewpoint, thereal Zariski sextics having no real points can be not only different but even non equivalent up toequisingular deformations; in fact, such “empty Zariski sextics” form two distinct deformationclasses (a twin pair of dual classes in the sense mentioned above).

Working with homology or cohomology we use by default Z-coefficients, dropping them fromthe notation. In the case of compact oriented even dimensional manifolds, especially in the caseof complex K3-surfaces, we identify the middle homology and cohomology lattices via Poincareduality up to omitting, with a slight abuse of notation, the duality operator.

Several other conventions related to lattices are introduced in the beginning of Section 3.

The symbol ⊡ is used to mark the end of a remark.

1.6. Acknowledgements. We thank G. Mikhalkin for sending to us his personal notes withthe figures illustrating construction of the Zariski sextics listed in [M]. This work was essentiallydone during the visits of the first author to Strasbourg University and partially during our visitsto MPIM (Bonn); we thank the both institutions for providing good working conditions. Wethank also an unknown referee for many valuable suggestions.

2. Zariski sextics

2.1. Generic projection in the complex setting. Let X be a non-singular cubic surface inP 3 defined by a homogeneous cubic polynomial f = f(x0, . . . , x3), ξ = [ξ0 : · · · : ξ3] a point inP 3(C)rX(C), and π = πX,ξ : X → P 2 the central projection from ξ. The critical set of π is the

curve B = BX,ξ = X ∩ Xξ traced on X by the polar quadric, Xξ, defined by fξ =∑3

i=0 ξifxi.

We call this curve B ⊂ X the rim-curve. The set of critical values, A = π(B), is called theapparent contour of X with respect to ξ. The lines passing through ξ and intersecting B traceon X another curve, B′, which we call the shadow contour, so that π−1(A) = B ∪B′.

We say that a point ξ is X-generic if Xξ is transverse to X , and B is transverse to the Hessian

plane Xξξ defined by fξξ =∑3

i,j=0 ξiξjfxixj. As is well known, the set of X-generic points ξ is


a non-empty Zariski-open subset of P 3 (one can find a detailed proof of the non-emptiness in[CF]).

Let us choose a coordinate system so that ξ turns into [0 : 0 : 0 : 1] and f appears in a depressedform (i.e., the quadratic in x3 term vanishes), f = x33 + px3 + q, where p and q are homogeneouspolynomials in x0, x1, x2 of degrees 2 and 3 respectively. We say that such a coordinate systemis associated with X and ξ (it is well-defined up to a coordinate change in the projection planeP 2 = [x0 : x1 : x2]).

2.1.1. Lemma. In a coordinate system associated with X and ξ,

(1) the polar quadric Xξ is 3x23 + p = 0,(2) the Hessian plane Xξξ is x3 = 0,(3) the apparent contour A is the sextic defined by the discriminant polynomial Df = 4p3 +

27q2 of f ,(4) the shadow-contour B′ is given by equations 3x23 + 4p = 0, x33 + px3 + q = 0.

Proof. Straightforward calculation.

2.1.2. Corollary. Assume that ξ is X-generic. Then,

(1) the rim curve B and the shadow-contour B′ are nonsingular and intersect each other atthe 6 points X ∩ Xξ ∩ Xξξ with multiplicity 2 (i.e., B and B′ have simple tangency atthese points);

(2) the apparent contour A is smooth except the 6 cuspidal points at π(X ∩Xξ ∩Xξξ).

2.1.3. Corollary. The point ξ is X-generic if and only if the conic p and cubic q intersecttransversely, and the sextic A has no other singularities except the 6 cusps at p = q = 0.

If a sextic has six cusps lying on a conic and no other singular points, we call it Zariski sextic.The following fact is also well known; its proof can be found in [Seg].

2.1.4. Proposition. A plane sextic is a Zariski sextic if an only if it is the apparent contourof a nonsingular cubic surface X with respect to an X-generic point. In particular, each Zariskisextic can be presented by equation 4p3 + 27q2, where p and q are homogeneous polynomials ofdegree 2 and 3 defining a conic and a cubic intersecting transversely. Such a presentation isunique up to rescaling (p, q) 7→ (t2p, t3q).

2.1.5. Corollary. The central projection correspondence that associates to a cubic surface x33+px3 + q = 0 and the point [0 : 0 : 0 : 1] the sextic 4p3 + 27q2 = 0 provides a homeomorphismbetween the space of projective classes of pairs (X, ξ), where X is a nonsingular cubic surfaceand a point ξ is X-generic, and the space of projective classes of Zariski sextics.

Proof. Bijectivity at the level of projective classes follows from Proposition 2.1.4. To constructcontinuous local inverse maps, it is sufficient for every Zariski sexticA to represent a neighborhoodof the PGL(3)-orbit of A as (G × E)/GA, where G = PGL(3), GA = AutA, and E is a finitedimensional vector space with a linear action of GA on it. Afterwards, it is enough to choosealong S = 1 × E ⊂ (G × E)/GA the family of equations 4p3s + 27q2s = 0, s ∈ S, in a way thatps, qs depend on s ∈ S continuously and equivariantly with respect to GA. The latter propertycan be achieved by simple averaging over the action of the group GA (recall that this group isfinite).

2.2. Generic projection in the real setting. From now on, we restrict our considerations tocubic surfaces X defined over R and points ξ in P 3(R)rX(R). However, we assume everywherethat the chosen point ξ is X-generic over C, in the sense of Section 2.1.

Since X, ξ are real, their Zariski sextic is real as well, and vice versa. More precisely, Propo-sition 2.1.4 and Corollary 2.1.5 imply the following.


2.2.1. Proposition. The central projection correspondence provides an isomorphism betweenthe three spaces:

(1) the set of real projective classes of real Zariski sextics,(2) the set of pairs (p, q) where p and q are real homogeneous polynomials of degree 2 and

3, respectively, considered up to simultaneous real projective transformations of variablesand satisfying two restrictions: first, the conic p = 0 and the cubic q = 0 intersecttransversally, and, second, the sextic 4p3 + 27q2 has no other singularities than 6 cusps;

(3) the set of real projective classes of pairs (X, ξ), where X stands for a nonsingular realcubic surface and ξ for a real X-generic point.

Remark. Recall that according to our convention, transversality is assumed at all the complexpoints, and that some of the six cusps of a real Zariski curve may be imaginary. ⊡

Let P± denote the region ±p 6 0 bounded in P 2(R) by our real conic p. Next, P 2(R) isdivided into two “halves” bounded by A(R), namely A± = [x] ∈ P 2(R) | ±Df (x) > 0. Notethat a Zariski sextic determines uniquely the sign of its degree 6 polynomial Df = 4p3 + 27q2,as well as the sign of p, and therefore determines the signs of the above regions, P± and A±.

At a neighborhood of a real cusp the curve divides the real plane into an “acute region”between the branches of the curves forming angle zero at the cusp, and the complementary“reflexive region”, so that we can speak on the acute and the reflexive sides of a curve at its realcusp. The definitions immediately imply the following.

2.2.2. Lemma.

(1) The real part of the apparent contour, A(R), lies entirely in P+.(2) The projection pR = p|X(R) is three-to-one over the interior points of A− and one-to-one

over the interior of A+.(3) The region A− lies entirely in P+ and bounds the real cusps of A from the acute side, see

Figure 1.

Figure 1. Zariski sextic on the right is obtained by a small perturbation 4(ǫp)3 +27q2 = 0 of the conic p = 0 and cubic q = 0 shown on the left; p < 0 inside the conicand ǫ > 0.

Note that one of the regions A± is orientable, and the other is not; let us denote themrespectively Ao and An, thinking of o and n as functions of (X, ξ) taking opposite values o, n ∈+,−. Then, in accordance with Lemma 2.2.2, o = + if and only if the projection pR = p|X(R)

is three-to-one over the interior points of the non-orientable component of P 2(R) r A(R). Dueto Proposition 2.1.4, these functions o and n can be also considered as functions of A.

The connected components of A(R) will be called ovals. An oval may have real cusps; it iscalled cuspidal in this case and smooth otherwise.

2.3. Restrictions on the arrangements of ovals and real cusps.

2.3.1. Lemma. Every cuspidal oval of a real Zariski sextic has an even number of cusps.


Proof. Every component of the real rim-curve B(R) is two-sided on X(R) (since the polar quadricXξ is two-sided), hence, null-homologous in P 3(R). Therefore, B(R) intersects the plane Xξξ

at an even number of points. But it is these points that are projected into the cusps of thecorresponding oval, see Corollary 2.1.2.

We denote by 2νi the number of the imaginary cusps of A, and by 2νr the number of realones, so that νi, νr > 0, and νi + νr = 3.

Every oval, O, of a Zariski sextic A is obviously null-homologous in P 2(R) and, thus, dividesthe latter into the interior of O (homeomorphic to a disc), and the exterior of O (homeomorphicto a Mobius band). Another oval lying in the interior (respectively, exterior) of O is called itsinternal (respectively, external) oval. If O has no internal ovals, then it is called empty oval. Anoval O is called ambient one with respect to its internal ovals.

A cusp on O may be directed towards the interior, or the exterior, of O and we call it aninward cusp, or outward cusp, respectively. The following statement follows directly from thedefinition of P± and A±.

2.3.2. Lemma. Any cusp of an oval O ⊂ A(R) is inward, if the region P+ lies inside theinterior of O, and outward, if it lies inside the exterior. All the real cusps are directed from A−

to A+.

2.3.3. Corollary. All the cusps on an oval of a real Zariski curve are alike: either all inward,or all outward.

2.3.4. Corollary. For any real Zariski sextic the following properties hold:

(1) there can not be more than one oval with inward cusps;(2) all the ovals in the exterior of an oval with inward cusps are smooth;(3) all the ovals in the interior of an oval with outward cusps are smooth;(4) if one of external ovals has an outward (inward) cusp, or one of internal ovals has an

inward (resp. outward) cusp, then o = − (resp. o = +).

2.3.5. Lemma. If some oval of a real Zariski sextic is non-empty, then no other its oval maycontain (a) an inward cusp, (b) more than two outward cusps.

Proof. (a) If an inward cusp is on an external oval, then we take a line passing through this cuspand a point inside an internal oval. If an inward cusp is on an internal oval, then it containsanother cusp and we take a line through both of them. In each of the cases there will be acontradiction to the Bezout theorem.

(b) If there is more than two outward cusps on an external oval, Ω, then we take a line passingthrough an internal oval and one of the four cusps on Ω chosen so that this line intersects Ω atsome other point (it is possible, since it contains > 2 cusps). If more then two, and thus, at leastfour outward cusps lie on an internal oval, Ω, then we can find a line passing through two ofthe cusps and intersecting Ω in at least one more point. This will also contradict to the Bezouttheorem.

2.3.6. Lemma. There cannot be more than one smooth empty oval bounding a disc containedin A−. In particular, there cannot be more than one smooth empty oval in the interior of an ovalwith inward cusps, as well as in the exterior of a non-empty oval with outward cusps.

Proof. The projection pR is 3-fold over A−, so, smoothness of the ovals implies that X(R) musthave spherical components projecting to the discs in A− bounded by these ovals. On the otherhand, X(R) has at most one spherical components.


2.4. The code of a Zariski sextic. The ambient differential-topological type ofA(R) in P 2(R),A being a Zariski sextic, is characterized by a certain code of A, which is defined as follows.

An oval with 2 6 2k 6 6 cusps has code 1k if the cusps are outward, and 1−k if the cuspsare inward; a smooth oval has code 1. The codes n and n1 are abbreviations for 1 ⊔ · · · ⊔ 1 and11 ⊔ · · · ⊔ 11 that denote a group of n empty ovals, which are all smooth, or respectively, all have2 outward cusps (recall that according to Corollary 2.3.4 groups of n > 2 ovals with inward cuspsare impossible). For an arrangement in which an oval O contains inside a set of ovals with thecode S, we use the code 1k〈S〉, 1−k〈S〉, or 1〈S〉 (depending on the number and the direction ofcusps on O). A few examples are shown on Figure 2.

3

131

1

1 1-1-1 < 2 >11-3

1

1

1

< 2 >1 1

2 1-1<1 >

1

1 1 1< < >>

Figure 2.

We may also ignore the cusps and describe the purely topological (i.e., class C0) arrangementsof ovals by dropping the subscripts from the codes. The result is called the simple code of A: itlooks like α if all the ovals are empty, like α⊔1〈β〉 if one of the ovals contains β ovals inside and αovals outside, or like 1〈1〈1〉〉, if there is a nest of three ovals. It is convenient to allow sometimesan alternative form of this code, and write α⊔ 1〈0〉 for an arrangement of α+1 empty ovals. Forthe empty set of ovals (A(R) = ∅), we use the code 0 and call it the null-code. The code 1〈1〈1〉〉is called the 3-nest code.

2.4.1. Lemma. Assume that a real sextic A has six cusps and no other singular points. Thenthe arrangement of its ovals has one of the following simple codes:

(1) α ⊔ 1〈β〉, 0 6 α, β 6 4, α+ β 6 4,(2) 0,(3) 1〈1〈1〉〉.

Proof. It follows from the Bezout theorem and Harnack’s estimate of the number of ovals (seeSubsection 2.5), like in the well-studied case of non-singular sextics, see, for example, Gudkov’ssurvey [G].

2.4.2. Lemma. A real sextic cannot have real singular points, if its real locus is a union ofthree nested ovals.

Proof. A line through a point inside the internal oval of the nest intersects such a sextic in 6 realpoints, which should be all non-singular not to contradict to the Bezout theorem.

2.5. The types I and II of (M-d)-curves. Recall that the Harnack inequality ℓ(A) 6 g(A)+1,which bounds the number of connected components, ℓ(A), of A(R) for a non-singular real curveA in terms of its genus, g(A), extends to singular irreducible curves as soon as one understandsby ℓ(A) and g(A) the number of components and the genus after normalization. Real curves


with ℓ(A) = g(A) + 1 are called M-curves. Let d(A) = g(A) + 1− ℓ(A). If d = d(A) > 0, then acurve is called an (M-d)-curve.

An irreducible curve A is said to be of type I if A(C) r A(R) is disconnected (i.e., has twocomponents), and of type II otherwise. The following is well-known.

2.5.1. Lemma. Assume that A is a real irreducible curve.

(1) If A is an M-curve, then it is of type I.(2) If A is an (M-d)-curve, where d is odd, then it is of type II.

If A is a real sextic whose real locus is a nest of three ovals, then A is of type I.

2.5.2. Corollary. A real Zariski sextic has at most 5 ovals. It is of type I if it has 5 ovals, andof type II if it has 4,2, or no ovals.

2.6. Reversion of Zariski sextics. Given an arrangement D ⊂ RP2 of ovals (smooth, orcuspidal), a point p ∈ RP2 rD, and a simple closed one-sided curve L ⊂ RP2 r (D ∪ p), onecan obtain another arrangement of ovals, D′ on the projective plane RP2

p,L obtained from the

given one by contracting L and blowing up p. If we pass to the isotopy classes of D and D′, onecan identify RP2

p,L with the initial RP2 and observe that the result of this operation depends

only on the choice of the component of RP2 rD containing p. An alternative description of thisoperation is to pick up an annulus neighborhood of D, D ⊂ N ⊂ RP2, N ∼= S1 × [−1, 1], and letD′ ⊂ RP2 be the image of D under the reversion mapping N → N , (x, t) 7→ (x,−t). We say thatD and D′ (considered up to isotopy) are in reverse position with respect to annulus N , or withrespect to point p, and call this operation reversion of D. An oval containing the point p inside,or equivalently, homologically non-trivial in N , will be called a principal oval of a reversion.

We say that D ⊂ N ∼= S1×[−1, 1] is trigonal with respect to N , if each segment x×[−1, 1] ⊂ Nintersects transverselyD at one or three points, except finitely many critical values of x for whichsegment x×[−1, 1] is tangent to D or contains a cusp whose tangent is transverse to this segment.The following observations are trivial.

2.6.1. Lemma. If an arrangement of ovals D is trigonal with respect to an annulus N , then ithas either one or three principal ovals. Moreover

(1) non-principal ovals are empty;(2) a non-principal oval cannot have inward cusps, and may have maximum two outward

cusps;(3) the ovals lying inside the principal oval, Ω, of D after reversion will lie outside the image

of Ω, and vice versa, the ones lying outside Ω will lie inside its image after reversion;(4) if D has three principal ovals, then there is no other ovals, there is no cusps on the ovals,

and D as well as its reversion D′ form a 3-nest arrangement 1〈1〈1〉〉.

We say that non-empty real Zariski sextics A and A′ are reversion partners if (a) A(R) istrigonal with respect to some annulus N ⊃ A(R) and A′(R) is obtained by reversion of A(R),(b) A and A′ are both of the same type, and (c) the signs o(A) and o(A′) are opposite.

Note furthermore that the definition of reversion implies that the cusps on the ambient ovalschange their shape after reversion: the inward cusps become outward and vice versa. And bycontrary, the cusps on the non-principal ovals do not change their shape.

2.6.2. Corollary. Assume that real Zariski sextics A and A′ are reversion partners. Then:

(1) if A has 3-nest code 1〈1〈1〉〉, then all three ovals are principal and A′ has the same code;(2) if an oval of A has an inward cusp, or more than two outward cusps, then it is principal;(3) if the simple code of A is α ⊔ 1〈β〉, β > 0, then the ambient oval must be principal and

A′ must have simple code β ⊔ 1〈α〉;(4) if the complete code of A is αk ⊔ 1m〈βn〉, β > 0, k, n,m ∈ Z (zero index here means that

it should be dropped), then the complete code of A′ is βn ⊔ 1−m〈αk〉.


Rule (4) in Corollary 2.6.2 can be extended to β = 0 in two cases: if by using 2.6.2(2) wecan determine which of the ovals is principal, or if all the ovals look alike. For instance, if thecomplete code of A is 1⊔ 1n, where n < 0, or n > 1, then A′ has complete code 1−n〈1〉. If A hascode n1, then A

′ has code 1−1〈(n− 1)1〉.

Remark. If A has, for instance, code 1 ⊔ 11 then the rules stated above do not give an answer,which of the ovals is principal, and so, one could question if the complete code of a reversionpartner of A is determined uniquely by the complete code of A? Is it possible for A to haveseveral reversion partners? In what follows we will prove that a real Zariski sextic A cannot havemore than one reversion partner. For example, the partner for A with the code 1 ⊔ 11 has code1〈11〉, and there is no Zariski sextic with the code 1−1〈1〉. ⊡

2.7. The relation between the topology of cubic surfaces and their Zariski sex-tics. Here, we consider a real nonsingular cubic surface X and one of its real Zariski sextics, A.Recall that the real locus X(R) of X may consist of two components, one homeomorphic to S2

and another to P 2(R). Otherwise, X(R) is homeomorphic to P 2(R) with h 6 3 handles. In par-ticular, X(R) is determined, up to homeomorphism, by its Euler characteristic χ(X(R)) = 1−2h,or equivalently, by h ∈ −1, 0, 1, 2, 3, where h = −1 corresponds to the case of disconnectedX(R).

2.7.1. Lemma. Assume that A has code α ⊔ 1〈β〉, so that d(A) = 4− α− β. Then

χ(X(R)) =

3 + 2(α− β)− 2νr = 4α+ 2d(A) + 2νi − 11 if o(A) = −,

1 + 2(β − α) − 2νr = 4β + 2d(A) + 2νi − 13 if o(A) = +.

In the case of null-code, χ(X(R)) equals 1 if o(A) = − and 3 if o(A) = +.In the case of 3-nest code, χ(X(R)) equals 3 if o(A) = − and 1 if o(A) = +.

Proof. As it follows from Corollary 2.1.2, χ(X(R)) = 3χ(A−) + χ(A+) − 2νr = 1 + 2χ(A−) −2νr.

2.7.2. Corollary. If A has code α ⊔ 1〈β〉, then

h(X) =

νr + β − α− 1 = 6− (2α+ d(A) + νi) if o(A) = −,

νr + α− β = 7− (2β + d(A) + νi) if o(A) = +.

If A has null-code, then h(X) = 0 if o(A) = −, and h(X) = −1 if o(A) = +.If A has 3-nest code, then h(X) = −1 if o(A) = −, and h(X) = 0 if o(A) = +.

The following observation is a kind of refinement of Lemma 2.3.6.

2.7.3. Lemma. If the disc bounded by a smooth empty oval of A(R) lies in A−, then h(X) = −1.In particular, for any A with code α ⊔ 1〈β〉, β > 0,

(1) if either some external oval or the ambient oval has an outward cusp and one of theexternal ovals is smooth, then νr = α− β;

(2) if either an internal oval has an outward cusp, or the ambient oval has an inward cuspand one of the internal ovals is smooth, then νr = β − α− 1.

Proof. Smoothness of an oval bounding a disc in A−, over which the projection X(R) → P (R)is three-to-one, implies that X(R) must contain a spherical component projecting to this disc,and thus, h(X) = −1. Assumption (1), as well as (2), guarantees that the smooth oval boundsa disc component of A−, and the conclusion h(X) = −1 is reformulated via Corollary 2.7.2.


2.7.4. Corollary. If all the ovals of A(R) are empty and o(A) = −, then either (a) all ovalshave outward cusps, or (b) one oval is smooth and each of the others has precisely two outwardcusps.

Proof. Since o(A) = −, the cusps are outward. By Lemma 2.3.6 there cannot be more than onesmooth oval, and letting β = 0 in Lemma 2.7.3(1), we see that in the presence of a smooth ovalthe others cannot have more than one pair of cusps.

2.8. Relation to the double covering cuspidal K3-surface Y . By taking double coveringπY : Y → P 2 ramified along Zariski sextic A, we obtain a K3-surface Y which has six cuspidalsingular points. There exist two liftings of the complex conjugation in P 2 to an involution inY ; they differ by the deck transformation of the covering and both are anti-holomorphic. Let uschoose and denote by conjY : Y → Y the one whose real locus, Y (R) = Fix(conjY ), is projectedby πY to An. We call such conjY the Mobius involution.

The group H2(Y )/Tors is a free abelian group of rank b2(Y ) = 22− 12 = 10. We consider theinvolution (conjY )∗ induced in H2(Y )/Tors and denote by r± the ranks of the ±1-eigengroups,x ∈ H2(Y )/Tors | (conjY )∗(x) = ±x.

2.8.1. Lemma. For any real Zariski sextic A we have χ(A−) = 1 + r+−r−2 . Specifically, if A

has code α ⊔ 1〈β〉, α, β > 0, so that d(A) = 4− (α+ β), then

r+ =(β − α) + 4 = 2β + d(A),

r− =(α− β) + 6 = 2α+ d(A) + 2.

If A has null code, then r+ = r− = d(A) = 5.If A has 3-nest code, then r+ = d(A) + 2 = 4 and r− = 6.

Proof. We have obviously r+ + r− = b2(X) = 10, and the Lefschetz fixed-point formula appliedto conjY yields r+ − r− = χ(Y (R)) − 2 = 2(χ(A−) − 1), where χ(A−) = β − α in the case ofcode α ⊔ 1〈β〉. For the null-code and 3-nest code one has χ(A−) = χ(RP2) = 1 and χ(A−) = 0respectively.

2.8.2. Corollary. For any real Zariski sextic the following three conditions are equivalent:

(1) the sextic has null-code;(2) r+ = d(A) = 5;(3) d(A) > r− − 2.

Next, we obtain the following relation between Y (R) and X(R), or more precisely, between

r± and h = 1−χ(X(R))2 .

2.8.3. Corollary. For any real Zariski sextic A with code α⊔1〈β〉 the following identities hold:

if o(A) = − then

r+ = νi + 2 + h,

r− = (4− νi) + 2 + (2 − h);

if o(A) = + then

r− = νi + 2 + (h + 1),

r+ = (4− νi) + 2 + (1 − h).

Proof. It follows from Lemma 2.8.1 and Corollary 2.7.2.


123456

Table 1A. Zariski sextics without a partner.

simple codes νr(A) o(A) complete codes types1〈4〉 0 − 1〈4〉 I1〈3〉 1 − 11〈3〉 II1〈2〉 1 + 1〈11 ⊔ 1〉 I1〈2〉 2 − 12〈2〉 II1〈1〉 3 − 13〈1〉 II1〈1〉 0 + 1〈1〉 II

2.9. Deformation classification statement. By the ID of a real Zariski sextic A we meanthe triple ID(A) = (complete code of A, type of A, o(A)), i.e., its complete code enhanced withtwo additional bits of information: the type (I or II), and the sign (+ or -).

Let us recall that two real Zariski sextics are said to be equivalent up to equivariant deforma-tion, or shortly belonging to the same deformation class if and only if they belong to the samecomponent of the space of real Zariski sextics.

2.9.1. Theorem. Each real Zariski sextic is determined up to equisingular deformation by itsID. The list of IDs of real Zariski sextics is given in Tables 1A-C.

The proof of Theorem 2.9.1 is one of our main goals. It is given in the very end of the paper,see Section 8.6.

3. Preliminaries on lattice theory

3.1. Even lattices and their discriminants. By a lattice we mean a free abelian group offinite rank endowed with a non-degenerate bilinear symmetric Z-valued pairing, called also theinner product of a lattice.

We denote by 〈n〉, n ∈ Z r 0, the lattice of rank 1 whose generator has square n, by U the

lattice of rank 2 defined by matrix

[0 11 0

], and by An (n > 1), Dn (n > 4), En (6 6 n 6 8) the

lattices generated by the corresponding negative definite root systems (the same notation, An,Dn, and En, is used also to refer to the corresponding types of simple singularities). Given twolattices L andM , we denote by L+M their direct sum (this notation does not lead to confusions,since non-direct sums of lattices are never considered). Notation nL, n > 1, stands for the directsum of n copies of L, while L(n), n ∈ Z r 0, denotes the result of rescaling the lattice L witha scale factor n, i.e., L(n) as a group coincides with L but the product of elements in L(n) is ntimes greater than in L. An isomorphism of lattices is indicated by writing L =M .

In this paper, we deal generally with even lattices, unless it is stated otherwise (although someof the techniques that we use or develop can be adapted to the case of odd lattices as well).Recall that a lattice L is called even, if x2 is even for any x ∈ L. When a lattice L is equippedwith a lattice involution c : L→ L, we introduce a c-twisted inner product 〈x, y〉c = x ·c(y), whichis obviously bilinear, symmetric and non-degenerate, and say that the pair (L, c) is of type I ifthe c-twisted product is even, and of type II otherwise.

Given a lattice, L, we denote by discrL its (finite abelian) discriminant group, L∗/L, whereL∗ = Hom(L,Z) is identified with a subgroup in L ⊗ Q by means of the lattice pairing. Thegroup discrL is endowed with the nondegenerate Q/Z-valued inner product [x][y] = xy mod Z,where x, y ∈ L∗ and [x], [y] stand for the cosets. Our assumption that the lattice L is evenendows discrL with a Q/2Z-valued quadratic refinement of the inner product. This refinement,qL : discrL → Q/2Z, is given by qL([x]) = x2 mod 2Z; it is called the discriminant form ofL. The relation qL([x] + [y]) = qL([x]) + qL([y]) + 2[x][y] implies that qL determines the innerproduct of discrL.


Zariski sextics with a partner

123

456789

1011121314151617181920

212223242526

27282930

31

Table 1B. The case of o(A) = −.

simple νr(A) complete typescodes codes

3 ⊔ 1〈1〉 3 31 ⊔ 1〈1〉 I2 ⊔ 1〈2〉 2 21 ⊔ 1〈2〉 I1 ⊔ 1〈3〉 1 11 ⊔ 1〈3〉 I

4 3 31 ⊔ 1 II2 ⊔ 1〈1〉 3 21 ⊔ 11〈1〉 II2 ⊔ 1〈1〉 2 21 ⊔ 1〈1〉 II1 ⊔ 1〈2〉 2 11 ⊔ 11〈2〉 II1 ⊔ 1〈2〉 1 11 ⊔ 1〈2〉 II1〈3〉 0 1〈3〉 II

3 3 31 I3 3 31 II3 2 21 ⊔ 1 II

1 ⊔ 1〈1〉 3 11 ⊔ 12〈1〉 II1 ⊔ 1〈1〉 2 11 ⊔ 11〈1〉 I1 ⊔ 1〈1〉 2 11 ⊔ 11〈1〉 II1 ⊔ 1〈1〉 1 11 ⊔ 1〈1〉 II1〈1〈1〉〉 0 1〈1〈1〉〉 I1〈2〉 1 11〈2〉 I1〈2〉 1 11〈2〉 II1〈2〉 0 1〈2〉 II

2 3 12 ⊔ 11 II2 2 21 II2 1 11 ⊔ 1 II

1〈1〉 2 12〈1〉 II1〈1〉 1 11〈1〉 II1〈1〉 0 1〈1〉 II

1 3 13 II1 2 12 II1 1 11 II1 0 1 II

0 0 0 II

Table 1C. The case of o(A) = +.

simple νr(A) complete typescodes codes

1 ⊔ 1〈3〉 3 1 ⊔ 1〈31〉 I2 ⊔ 1〈2〉 2 2 ⊔ 1〈21〉 I3 ⊔ 1〈1〉 1 3 ⊔ 1〈11〉 I

1〈3〉 3 1〈31〉 II1 ⊔ 1〈2〉 3 1 ⊔ 1−1〈21〉 II1 ⊔ 1〈2〉 2 1 ⊔ 1〈21〉 II2 ⊔ 1〈1〉 2 2 ⊔ 1−1〈11〉 II2 ⊔ 1〈1〉 1 2 ⊔ 1〈11〉 II

4 0 4 II

1〈2〉 3 1−1〈21〉 I1〈2〉 3 1−1〈21〉 II1〈2〉 2 1〈21〉 II

1 ⊔ 1〈1〉 3 1 ⊔ 1−2〈11〉 II1 ⊔ 1〈1〉 2 1 ⊔ 1−1〈11〉 I1 ⊔ 1〈1〉 2 1 ⊔ 1−1〈11〉 II1 ⊔ 1〈1〉 1 1 ⊔ 1〈11〉 II1〈1〈1〉〉 0 1〈1〈1〉〉 I

3 1 1−1 ⊔ 2 I3 1 1−1 ⊔ 2 II3 0 3 II

1〈1〉 3 1−2〈11〉 II1〈1〉 2 1−1〈11〉 II1〈1〉 1 1〈11〉 II2 2 1−2 ⊔ 1 II2 1 1−1 ⊔ 1 II2 0 2 II

1 3 1−3 II1 2 1−2 II1 1 1−1 II1 0 1 II

0 0 0 II

The pair (discrL, qL) is called the discriminant of L, and when it does not lead to a confusionis denoted simply by discrL or qL.

3.2. Groups with inner products and quadratic refinements. A finite abelian group Gendowed with a non-degenerate symmetric bilinear form G × G → Q/Z will be called a finiteinner product group. We denote by ab the inner product of elements a, b ∈ G and use notation〈mn〉 (with coprime m and n) for a finite inner product cyclic group Z/n that has a2 = m

n∈ Q/Z

for one of generators a ∈ Z/n.


If a finite inner product group G is endowed additionally with a quadratic refinement, q : G→Q/2Z, then G will be called an enhanced group. By definition, q must be related to the innerproduct as follows: for all a, b ∈ G, n ∈ Z,

(1) q(a+ b) = q(a) + q(b) + 2ab,(2) q(na) = n2

q(a).

These relations imply q(−a) = q(a), q(a) = a2 mod Z, and q(2a) = 4a2 mod 2Z (note that4a2 is well-defined modulo 4 and, thus, modulo 2). The latter relation shows that q(a) is defineduniquely by the inner product as soon as a is divisible by 2. In the context of enhanced groups,notation 〈m

n〉 (with coprime m and n) has a meaning that q(a) = m

n∈ Q/2Z for a generator

a ∈ Z/n. Note that in this case either m or n must be even, since q((n+ 1)a) = q(a) if and onlyif m

n((n+ 1)2 − 1) is even.

3.2.1. Lemma. Any finite inner product group G of odd order has a canonical quadratic refine-ment and, thus, is an enhanced group.

Proof. Let q(a) = 4(a2 )2 mod 2Z, then all the required properties are satisfied.

3.2.2. Example. A finite inner product group 〈13 〉 being enhanced takes notation 〈− 23 〉 accord-

ing to our conventions. An enhanced group 〈16 〉 splits into a direct (orthogonal) sum 〈− 12 〉+ 〈23 〉.

In general, the set of quadratic refinements of a given inner product in a group G form anaffine space over Hom(G,Z/2Z).

We say that a subgroupK ⊂ G of a finite inner product group G is non-degenerate if its kernelx ∈ K |xK = 0 is trivial. It is straightforward to check the following.

3.2.3. Lemma. If K is a non-degenerate subgroup of a finite inner product group G, then Ksplits out as a direct summand, G = K +K⊥.

3.3. The p-components. By a p-group, where p is prime, we mean a finite abelian group G,such that the order, ord(x), of any element x ∈ G is a power of p. Note that any p-group G canbe presented as a direct sum of cyclic subgroups of the form Z/pk, k > 1, and that the numberof summands is an invariant of G independent of a decomposition; it is called the rank of G. Iford(x) = p for all non-identity x ∈ G, then G is called an elementary p-group; such a group Gcan be viewed as a vector space over Z/p.

Any finite abelian group G splits into a direct sum of its maximal p-subgroups Gp called primecomponents, or p-components of G. The p-primary component of G is non-trivial if and only ifp divides the order of G and coincides with the subgroup formed in G by elements whose orderis a power of p. With respect to any inner product in G the subgroups Gp must be orthogonal,which reduces studying of inner products to the case of p-groups.

3.3.1. Lemma. Assume that G is a finite inner product group (for instance, an enhancedgroup).

(1) If x ∈ G has order n, then x2 ∈ 1nZ/Z (respectively, q(x) ∈ 1

nZ/2Z).

(2) Any two prime components, Gp1 and Gp2 , p1 6= p2, are orthogonal with respect to theinner product in G.

Proof. If x, y ∈ discrL and x has order n, then xy ∈ 1nZ/Z and x is orthogonal to ny, since

nxy = x(ny) = (nx)y = 0 ∈ Q/Z. This implies both (1) and (2), since q(x) = x2 mod Z.

3.4. The discriminant p-ranks of lattices. In a particular case of a lattice L (which canbe odd here) and G = discrL (viewed here only as a group), we call the prime componentsGp the discriminant p-components and their ranks the discriminant p-ranks of L. We denotethem discrp L and rp = rp(L), respectively. The primes p for which discrp L 6= 0 (i.e., the primedivisors of | discrL|) are called the discriminant factors of L. A discriminant factor p is said to


be elementary if discrp L is an elementary p-group, we say also that L is p-elementary in thiscase. A lattice L is said to be divisible by p if xy is divisible by p for all x, y ∈ L, or equivalently,if L′ = L( 1

p) is a lattice.

Consider an endomorphism mp : discrL → discrL, mp(x) = px, then its kernel Kp ⊂ discrLis an elementary p-group of rank rp. Let L∗

p = ψ−1(Kp) ⊂ L∗ be the pull-back with respect tothe projection ψ : L∗ → discrL = L∗/L.

3.4.1. Proposition. For any lattice L and prime p, the p-rank rp of L is not greater than therank r of L. Moreover, L is divisible by p if and only if rp = r. In the latter case the followingproperties hold:

(1) there is a canonical exact sequence 0 → discrL′ f−→ discrL → L∗/pL∗ → 0, whereL∗/pL∗ = (Z/p)r, and the restriction of f yields a group isomorphism discrq L

′ = discrq Lfor any prime q 6= p, as well as an isomorphism between discrp L

′ and the subgroupp discrp L ⊂ discrp L;

(2) the group isomorphism discrL′ → f(discrL′) ⊂ discrL identifies the discriminant innerproduct in discrL restricted to f(discrL′) with the discriminant inner product in discrL′

multiplied by p, and if L′ is even, then qL|f(discrL′) is identified with pqL′ .

Proof. Note that Kp = (Z/p)rp ⊂ discrL is the maximal subgroup of exponent p in discrp L (andthus, in discrL). As it follows from definition, pL∗

p ⊂ L, and thus,

Kp = L∗p/L ⊂ L∗

p/pL∗p,

where the latter group is isomorphic to (Z/p)r, since L∗p is a free abelian group of rank r (as

it contains L). Thus, rp 6 r. In the case of equality rp = r, we have Kp = L∗p/pL

∗p, that is

L = pL∗p. So, for any x, y ∈ L, x( 1

py) ∈ Z, since 1

py ∈ L∗

p ⊂ L∗, and thus, xy is divisible by p,

and so L is divisibly by p. Conversely, if L is divisibly by p, then xp∈ L∗

p for any x ∈ L, and so

L = pL∗p and Kp = L∗

p/pL∗p has rank rp = r.

The exact sequence in (1), with its properties, follow from the observation that under ouridentification of L′ with L as a group, the dual (L′)∗ of L′ is identified with pL∗.

Remark. Given a free abelian group M of rank m, and its subgroup L ⊂ M of the same rank,one can find a basis e1, . . . , em of M , such that L is spanned by some multiples k1e1, . . . , kmem,ki > 1, and ki divides ki+1 for i = 1, . . . ,m − 1. This yields a direct sum decompositionM/L = Z/k1 + · · ·+ Z/km. If we apply it to L ⊂ L∗, then we get another proof of Proposition3.4.1. ⊡

3.4.2. Lemma. For every even lattice L, we have r2(L) = r(L) mod 2.

Proof. In accordance with the definition of the 2-rank, r2 is equal to the rank of the subgroup ofdiscrL that is formed by elements of order 2. In its turn, the rank of this subgroup is equal tothe rank of discr(L)/2 discr(L), and thus to the rank of the radical of the mod 2 reduction of L(as a quadratic space). This reduction is even, since L is even. Therefore, r(L) − r2(L) as therank of a non degenerate even Z/2-valued form is even.

3.5. The Brown invariant. As is known, for any enhanced group (G, q), the Gaussian sum

G(q) =∑

x∈G eπiq(x) has absolute value

√|G|, whereas G(q)√

|G|belongs to the group µ8 of eight’s

roots of 1. In modern terminology one speaks on the (generalized) Brown invariant, Br(q) ∈Z/8 ∼= µ8. This Gaussian sum appears in Van der Blij’s famous formula that relates, in the caseof even lattices and their discriminants, the argument of G(q) with the signature of the lattice.


3.5.1. Theorem. (Van der Blij [VdB]) If L is an even lattice, then Br(qL) is equal to the mod8 residue of the signature σ(L) of L.

Note that the definition via Gauss sums immediately implies additivity,

Br(q1 ⊕ q2) = Br(q1) + Br(q2).

Speaking on even lattices L, we let Brp(L) = Br(qL|discrp(L)) and use an alternative notationBr(L) for Br(qL) = σ(L) mod 8 (since in certain formulas it is more instructive and convenientthan σ(L) mod 8). Then Lemma 3.3.1 and the above additivity formula imply the followingstatement.

3.5.2. Proposition. For any even lattice L, we have Br(L) =∑

prime p Brp(L).

3.6. Elementary enhanced 2-groups. Assume that G is an enhanced 2-group with the qua-dratic refinement q. The finite inner product of G is said to be even if x2 = 0 for all x ∈ G, orin terms of q, if the values of q(x) are integer (and thus, q(x) ∈ Z/2Z). Otherwise, the innerproduct is called odd. We encode this parity by putting δ2(q) = 0 in the even case, and δ2(q) = 1otherwise. For each 2-elementary lattice L, we define its discriminant parity δ2(L) ∈ Z/2 byletting δ2(L) = δ2(qL).

3.6.1. Example. The group Z/2 = [0], [1] has a unique inner product, [1][1] = 12 , [∗][0] =

[0][∗] = 0, but two possible quadratic refinements, q([1]) = ± 12 , q([0]) = 0, denoted in accordance

with our convention in Section 3.2 by 〈± 12 〉. Note that an even enhanced 2-group cannot contain

〈± 12 〉 as a direct summand.

Observing that discr(〈±2〉) = 〈± 12 〉 we obtain δ2(〈±2〉) = 1 and Br(〈±2〉) = ±1.

3.6.2. Example. The discriminant groups of lattices U(2) and D4 are both isomorphic to

G = Z/2 + Z/2, with the inner product

[0 1

212 0

]. The quadratic refinements are qU(2)(1, 0) =

qU(2)(0, 1) = 0, qU(2)(1, 1) = 1, and qD4(1, 1) = qD4(1, 0) = qD4(0, 1) = 1. We denote the cor-responding enhanced groups by U2 and V2 respectively. Here we have δ2(U(2)) = δ2(D4) = 0,whereas Br(U(2)) = 0 but Br(D4) = 4.

The following statement is a straightforward consequence of well known ”uniqueness” results,see [W1] (cf., [GM]).

3.6.3. Theorem. Any elementary enhanced 2-group (G, q) is characterized up to isomorphismby its rank, the Brown invariant, and the parity. The only non-split enhanced 2-groups are 〈± 1

2 〉,U2, and V2. Moreover:

(1) If δ2(q) = 0, then G = aU2 + bV2, for some a, b > 0. In particular, the rank of G is evenand Br(q) = aBr(U2) + bBr(V2) = 4b mod 8 is divisible by 4.

(2) aU2 + bV2 is isomorphic to a′U2 + b′V2 if and only if a+ b = a′ + b′ and a = a′ mod 2.(3) If δ2(q) = 1, then G = a〈12 〉 + b〈− 1

2 〉, for some a, b > 0, a + b = r2. In particular,Br(q) = a− b mod 8.

(4) a〈12 〉 + b〈− 12 〉 is isomorphic to a′〈12 〉 + b′〈− 1

2 〉 if and only if a + b = a′ + b′ and a = a′

mod 4.

If G is a finite inner product elementary 2-group, then the map x 7→ x2 is a homomorphismG→ 1

2Z/Z = Z/2 and, therefore, there exists one and only one element v ∈ G such that vx = x2

for all x ∈ G; this v is called the characteristic element of G.


3.6.4. Lemma. Assume that (G, q) is an elementary enhanced 2-group and v ∈ G such thatq(v) 6= 0. Then:

(1) The orthogonal complement v⊥ is an enhanced subgroup of G.(2) v⊥ is even if and only if v is the characteristic element of G.(3) If v is characteristic, then 2q(v) ∈ Z/4Z equals to the mod 4-residue of Br(q) ∈ Z/8.(4) If q(v) = ± 1

2 , then the Brown invariant Br〈v〉 of the subgroup 〈v〉 ⊂ G spanned by v isrespectively ±1.

Proof. Item (4) is already seen in Example 3.6.1. Item (1) follows from non-degeneracy of q|v⊥ ,which is due to q(v) 6= 0. Item (2) follows from q(x) = xv mod Z, by the definition of thecharacteristic element v. To prove item (3) it is sufficient, due to additivity of q(v) and Br withrespect to the direct sums and the classification given in Theorem 3.6.3, to check the requiredrelation on the four non-split enhanced groups 〈12 〉, 〈− 1

2 〉, U2, and V2 (cf., Examples 3.6.1 and3.6.2).

3.7. Elementary 3-groups with an inner product. The group G = Z/3 = [0], [1], [2]admits two different inner product structures, 〈13 〉, and 〈− 1

3 〉. For the first one, [1][1] = [2][2] = 13 ,

and for the other, [1][1] = [2][2] = − 13 . The quadratic refinements take values q([1]) = q([2]) = − 2

3

and q([1]) = q([2]) = 23 , respectively (cf. Example 3.2.2); that is why to indicate that we deal

with these enhanced structures we use notation 〈− 23 〉 and 〈23 〉.

3.7.1. Example. The discriminant of the lattice A2 is 〈− 23 〉, while discrA2(−1) and discrE6

are both isomorphic to 〈23 〉. It is easy to check also that discr3〈±6〉 = discr3 A2(±2) = 〈± 23 〉.

The statements of the following Lemma are well known and can be easily extracted, forexample, from [W1].

3.7.2. Lemma.

(1) Any finite inner product 3-group is isomorphic to a〈23 〉+b〈− 23 〉, for some a, b > 0, a+b =

r3.(2) Br(a〈23 〉+ b〈− 2

3 〉) = 2(a− b) mod 8.

(3) a〈23 〉 + b〈− 23 〉 is isomorphic to a′〈23 〉 + b′〈− 2

3 〉 if and only if a + b = a′ + b′ and a = a′

mod 2. In the other words, such inner product groups are isomorphic if and only if theirranks and Brown invariants coincide.

3.7.3. Corollary. Finite inner product 3-groups (G, q) are characterized up to isomorphism bypairs (a, b), where a ∈ 0, 1 and b > 0, such that G = a〈23 〉+ b〈− 2

3 〉.

Theorem 3.5.1 and Proposition 3.5.2 with Lemma 3.7.2 imply the following.

3.7.4. Corollary. Assume that a lattice L is even and has only discriminant factors 2 and 3,which are both elementary. Then

Br2(L) + Br3(L) = σ(L) mod 8,

where Br3(L) = 2(a− b) mod 8 if discr3 L = a〈23 〉+ b〈− 23 〉.

3.8. Extensions of lattices. Consider an even lattice L and its extension M ⊃ L, that isanother even lattice of finite index [M : L]. By means of the lattice pairing every such M iscanonically embedded in L ⊗ Q, and we consider as equivalent the extensions having the sameimage in L ⊗ Q. Furthermore, L ⊂ M ⊂ M∗ ⊂ L∗ and the subgroup H = M/L ⊂ discrLis isotropic, i.e., qL vanishes on H . Conversely, for any isotropic subgroup H ⊂ discrL, thepreimage, LH ⊂ L∗, of H under the quotient map L∗ → L∗/L is an even lattice. This impliesthe following (where 3.8.1(1), 3.8.2, and 3.8.3 are shown in [N1], while 3.8.1(2) follows from thevan der Blij Theorem 3.5.1).


3.8.1. Lemma. If M is an extension of L associated with an isotropic subgroup H of discrL,then:

(1) discrM = H⊥/H, where H⊥ = M∗/L = x ∈ discrL |xH = 0 is the orthogonalcomplement of H;

(2) Br(L) = Br(M).

3.8.2. Lemma. For any even lattice L the correspondence between the isotropic subgroups ofH ⊂ discrL and the extensions of L is one-to-one, and discrLH = H⊥/H.

3.8.3. Lemma. Let M1,M2 be two extensions of L associated with isotropic subgroups H1, H2

of discrL. An automorphism f : L → L can be extended to an isomorphism M1 → M2 if andonly if the induced automorphism of discrL maps isomorphically H1 onto H2.

3.9. Gluing of lattices. Consider a pair of even lattices L1, L2 and subgroups Ki ⊂ discrLi,i = 1, 2. We say that φ : K1 → K2 is an anti-isomorphism if it is a group isomorphism such thatqL1(x) = −qL2(φ(x)) for all x ∈ K1. For any such an anti-isomorphism φ the graph-subgroupHφ = x+φ(x) |x ∈ K1 ⊂ K1+K2 is isotropic in discr(L1+L2) = discrL1+discrL2 and thusdefines an extension of L1 + L2. We denote this extension by L1 +φ L2 and say that the latteris the result of gluing L1 with L2 along φ.

Recall that a sublattice L1 ⊂ L of a lattice L is called primitive, if the group L/L1 containsno torsion. It is trivial for instance, that the orthogonal complement L2 = L⊥

1 = x ∈ L |xy =0 for all y ∈ L1 of any sublattice L1 ⊂ L is primitive, and primitivity of L1 is equivalent to thatL1 = L⊥

2 . Note that for each i = 1, 2 the image of L by the orthogonal projection to Li ⊗ Q iscontained in L∗

i ⊂ L⊗Q and the kernel of the composition L→ L∗i → L∗

i /Li is L1 +L2, so thatthere appear two well defined induced monomorphisms pi : L/(L1 + L2) → discrLi. Thus, weget the following (see [N1]).

3.9.1. Proposition. For any gluing L = L1 +φ L2, the lattices L1 and L2 are orthogonalcomplements of each other, and thus primitive, in L. Conversely, if two even sublattices L1, L2

of a lattice L are orthogonal complements of each other, then:

(1) they determine canonically subgroups Hi ⊂ discrLi, i = 1, 2, and an anti-isomorphismφ : H1 → H2, so that L can be identified with L1 +φ L2 by an isomorphism identical onL1 and L2;

(2) the above subgroups Hi are nothing but the images, pi(L/(L1+L2)), and φ(p1(x)) = p2(x)for all x ∈ L/(L1 + L2).

3.9.2. Corollary. Gluing L = L1 +φ L2 is an even lattice with discrL = H⊥φ /Hφ.

Corollary 3.9.2 together with Lemma 3.2.3 imply the following.

3.9.3. Proposition. Let L1 +φ L2 be a gluing along φ : K1 → K2. If Ki ⊂ discrLi, i = 1, 2,are non-degenerate then discr(L1 +φ L2) = K⊥

1 +K⊥2 .

Consider two gluings: L(1)1 +φ(1) L

(1)2 along φ(1) : K

(1)1 → K

(1)2 and L

(2)1 +φ(2) L

(2)2 along φ(2) :

K(2)1 → K

(2)2 . We say that homomorphisms fi : L

(1)i → L

(2)i , i = 1, 2, are (φ(1), φ(2))-compatible,

if the induced homomorphisms fdiscri : discrL

(1)i → discrL

(2)i restricted to K

(j)i commute with

φ(j):

K(1)1

fdiscr1−−−−→ K

(2)1yφ(1) φ(2)

y

K(1)2

fdiscr2−−−−→ K

(2)2 .

Lemma 3.8.3 immediately implies the following.


3.9.4. Lemma. Homomorphisms fi : L(1)i → L

(2)i , i = 1, 2, can be extended in a unique way to

a homomorphism f : L(1)1 +φ(1) L

(1)2 → L

(2)1 +φ(2) L

(2)2 if and only if fi are (φ(1), φ(2))-compatible

If fi are (φ(1), φ(2))-compatible isomorphisms and fdiscri are isomorphisms, then f is also an

isomorphism.

3.10. The orthogonal complement of ±2-elements. Primitive lattice elements v ∈ L withv2 = n are called n-elements. For n 6= 0, the sublattice Zv generated by such a v is isomorphicto 〈n〉 and its orthogonal complement is denoted by v⊥ or Lv.

For a given n-element v in an even lattice L we have L = 〈n〉 +φ Lv with φ : Kv → Kv,

Kv ⊂ discr〈n〉 and Kv ⊂ discrLv. If n = ±2, there are two cases: both Kv and Kv are trivial,or Kv = discr〈±2〉 = 〈± 1

2 〉 and Kv = 〈∓ 12 〉. We say that v is even in the first case, and that v

is odd in the second.

3.10.1. Lemma. For any ±2-element v in an even lattice L the conditions below are equivalent:

(1) v is even,(2) discrL = 〈± 1

2 〉+ discrLv,(3) L = Zv + Lv,(4) the product vx is even for all x ∈ L,(5) v

2 ∈ L⊗Q lies in L∗ and its coset [ v2 ] is a non-trivial element of discrL.

Proof. Equivalences (1) ↔ (2) ↔ (3) and (4) ↔ (5) are evident. The remaining equivalencefollows from the orthogonal projection formula projv x = vx

vvv.

3.10.2. Lemma. For any odd ±2-element v in an even lattice L, the complementary discrimi-nant discrLv is isomorphic to discrL+ 〈∓ 1

2 〉.

Proof. Since Kv = 〈∓ 12 〉 is non-degenerate, it splits off as a direct summand of discrLv by

Lemma 3.2.3.

If an even lattice L is 2-elementary, then its even ±2-elements are subdivided into two species:ordinary even elements and Wu elements. By definition, an even element v is a Wu element if [ v2 ]is the characteristic element of discr2 L, i.e., qL(x) = x[ v2 ] mod Z for all x ∈ discr2 L; otherwisev is ordinary.

3.10.3. Lemma. An even ±2-element v ∈ L in a 2-elementary even lattice L is a Wu elementif and only if the complementary discriminant discrLv is even, i.e., qL(x) ∈ Z/2Z ⊂ Q/2Z forall x ∈ discrLv.

Proof. Follows from Lemma 3.10.1(3) and Lemma 3.6.4.

3.11. Involutions via gluing. Consider an even lattice L, a lattice involution c : L→ L, andits eigenlattices L± = x ∈ L | c(x) = ±x. Note that L/(L+ + L−) is an elementary 2-group,since 2x = (x + c(x)) + (x − c(x)), x ± c(x) ∈ L±, for all x ∈ L. Let r2(L, c) denote the 2-rank of L/(L+ + L−). Consider the projection L → L∗

± sending x ∈ L to y 7→ xy, y ∈ L±,denote by q± : L → discrL± its composition with the quotient map L∗

± 7→ discrL± and putK± = q±(L) ⊂ discrL±. Clearly, each of q± induces a group isomorphism between L/(L++L−)and K±; in particular, they give rise to a canonical isomorphism φ : K+ → K−.

The following proposition showing how c can be described in terms of 2-elementary subgroupsof discrL± is essentially Proposition 1.2.1 in [N3].

3.11.1. Proposition.

(1) A lattice involution c : L→ L yields a presentation of L as a result of gluing L = L+ +φ

L− of its eigenlattices along an anti-isomorphism φ : K+ → K− between 2-elementarysubgroups K± ⊂ discrL±.


(2) Conversely, if a lattice L is glued from even lattices, L = L+ +φ L−, along an anti-isomorphism φ : K+ → K− between 2-elementary subgroups K± ⊂ discrL±, then thereexists a lattice involution c : L→ L, for which L± are the (±1)-eigenlattices.

3.11.2. Proposition. Assume that L is an even lattice with discr2 L = 〈± 12 〉 and c : L → L is

a lattice involution. Then the eigenlattices L± of c are 2-elementary, and |r2(L+)− r2(L−)| = 1.

Proof. Let us first extend c to the lattice L′ = L+φ 〈∓2〉, where φ : discr2 L→ discr〈∓2〉 = 〈∓ 12 〉

is the anti-isomorphism. Namely, choosing one of the two generators, h ∈ 〈∓2〉 we consider theinvolution defined in 〈∓2〉 by h 7→ εh, where ε ∈ +,− (i.e., the identity or the anti-identity).Lemma 3.9.4 allows to glue the involution c with the latter to obtain cε : L

′ → L′, which is aninvolution as well. Since discr2 L

′ = 0, Propositions 3.11.1 and 3.9.3 implies that the eigenlatticesL′± of cε are glued into L′ along ψ : discr2 L

′+ → discr2 L

′−, and that discr2 L

′± are 2-elementary.

On the other hand, h ∈ L′ε, thus L−ε = L′

−ε, and applying Lemmas 3.10.1(2) and 3.10.2 we

conclude that either discrLε = discrLε + 〈± 12 〉 (if h is odd) or discrLε + 〈∓ 1

2 〉 = discrLε (if h iseven).

Given a lattice involution c on an even lattice L, denote by c2 the involution induced inL2 = L⊗ Z/2 by c and put

r2(L2, c2) = rank(L2/Lc22 ), where Lc2

2 = x ∈ L2 | c(x) = x.

3.11.3. Proposition. If L has odd discriminant, then

(1) δ2(L+) = 0 if and only if (L, c) is of type I.(2) r2(L, c) = r2(L2, c2) = r2(L±).

Proof. The preimage of Lc22 under the reduction homomorphism L→ L⊗Z/2 = L2 is L+ ⊕L−,

which implies r2(L, c) = r2(L2, c2). Each element a ∈ discr2 L+ is represented by an element ofthe form 1

2 (x ± cx), x ∈ L, which implies r2(L, c) = r2(L±). To prove (1), it remains to notice

that q(a) = (12 (x+ cx))2 = 12 (x

2 + x · cx) = 12x · cx ∈ Q/Z.

3.12. Stability. An even lattice L is called stable if any other even lattice L′ having the same in-ertia indices, r+(L

′) = r+(L) and r−(L′) = r−(L), and isometric discriminants, (discrL′, qL′) =

(discrL, qL), is isomorphic to L (such a stability is equivalent to what is also phrased as ”unique-ness in its genus”, see [N3]).

Let us call an even lattice L epistable (respectively, p-epistable) if any automorphism of discrL(respectively, of discrp L) is induced by some automorphism of L.

The following Nikulin’s criterion shows that such complications as non-stability or non-epistabilitymay happen only if the lattice has the extremal or next to extremal value, r or r− 1, (cf. Propo-sition 3.4.1) of the ranks rp for some prime p.

3.12.1. Theorem. (Nikulin [N1]) Assume that a lattice L of rank r is even, indefinite, and theranks rp satisfy the following conditions:

(1) rp 6 r − 2 for all primes p 6= 2;(2) if r2 = r, then discr2 L contains U2 or V2 as a direct summand.

Then L is both stable and epistable (and in particular, p-epistable for all p).

Note that, as it follows from Theorem 3.6.3, the condition (2) is satisfied, if discr2 L is 2-elementary and r2 = r > 2, so the condition (2) requires analysis only for r2 = r = 2. Note alsothat any even lattice L of rank 1 is stable, since L = 〈2n〉, where n is determined by discrL.

R. Miranda and D. Morrison [MM1], [MM2] developed further Theorem 3.12.1 and gave anecessary and sufficient criterion of stability and epistability, which is in the special case of ourinterest looks as follows.


3.12.2. Proposition. Suppose that L is an indefinite even lattice of rank r > 3, which has onlydiscriminant factors 2 and 3, and the latter ones are elementary. Then L is stable and epistable,except possibly the case r2 = r3 = r.

3.13. Different involutions with the same eigenlattices. Consider a pair of even latticesT 1 and T 2 with lattice involutions, ci : T

i → T i, whose eigenlattices are isomorphic, T 1± = T 2

±.

By Proposition 3.11.1, T i = T i+ +φi T i

−, where φi : Ki

+ → Ki−, i = 1, 2, are anti-isomorphisms

between elementary 2-groups Ki± ⊂ discr2 T

i±. We say that involutions c1 and c2 are conjugate

via some isomorphism f : T 1 → T 2 if f c1 = c2 f . The description of isomorphisms via gluingin Lemma 3.9.4 yields easily the following.

3.13.1. Proposition. The following conditions are equivalent:

(1) c1 and c2 are conjugate via some isomorphism f : T 1 → T 2;(2) there exists an isomorphism f : T 1 → T 2 which maps isomorphically T 1

± onto T 2±;

(3) there exist isomorphisms f± : T 1± → T 2

± such that the induced ones, fdiscr2± : discr2 T

1± →

discr2 T2±, map isomorphically K1

± onto K2± so that the following diagram commutes.

K1+

fdiscr2+−−−−→ K2

+

φ1

y φ2

y

K1−

fdiscr2−−−−−→ K2

−

The next Proposition gives a criterion for involutions to be conjugate under additional as-sumption that discr2 T

i is Z/2 as a group.

3.13.2. Proposition. Assume that ci : Ti → T i, i = 1, 2, are lattice involutions, whose eigen-

lattices T i± are respectively isomorphic, namely T 1

± = T 2±. Assume, in addition, that r2(T

i+) <

r2(Ti−) and discr2 T

i = 〈ε 12 〉,ε ∈ +,−, for each i = 1, 2. Then c1 and c2 are conjugate via

some isomorphism f : T 1 → T 2 if any one of the following conditions is satisfied:

(1) the lattice T 1− is 2-epistable;

(2) the lattice T 1+ is 2-epistable and Aut(T 1

−) acts transitively on the subgroups of discr2 T1−

anti-isomorphic to discr2 T1+.

Proof. By Proposition 3.11.1, we have T i = T i+ +φi T i

− with φi : Ki+ → Ki

−, i = 1, 2. According

to Proposition 3.11.2, T i± are 2-elementary, Ki

+ = T i+ (since r2(T

i+) < r2(T

i−) and r2(T

i) = 1),

and Ki− ⊂ T i

− is a subgroup of corank 1. Note also that the orthogonal complements of Ki−

in discr2 Ti− are isomorphic, since the enhanced-group structures on Z/2 are determined by the

Brown invariant. Thus, K1− is sent to K2

− by some isomorphism fdiscr2− : discr2 T

1− → discr2 T

2−.

Moreover, we can choose it so that the diagram in Proposition 3.13.1(3) commutes. Namely, inthe case (1), the 2-epistability of T 1

−∼= T 2

− implies existence of an isomorphism f− : T 1− → T 2

−

inducing fdiscr2− . Then, Proposition 3.13.1 implies that c1 and c2 are conjugate via f defined

by f+ = id and f− constructed above. In the case (2), we can find f− : T 1− → T 2

− such thatthe induced map in discr2(T

1−) sends K1

− to K2−. Then we can use the epistability of T 1

+ toconstruct f+ : T 1

+ → T 2+ which is compatible with f−, that is the diagram like in Proposition

3.13.1 commutes, and we again conclude that c1 and c2 are conjugate.

4. Topology and arithmetics of the covering K3-surfaces

4.1. Covering K3 after desingularization. In addition to the cuspidal K3-surface Y intro-duced in Section 2.8 and obtained by taking the double covering of P 2 ramified along a Zariski


curve A, we take also into consideration the non-singular K3 surface Y obtained by the minimal

resolution of the six cusps of Y . Note that Y inherits from Y a pair of complex conjugations that

differ by the deck transformation of the double covering Y → P 2 of the plane blown-up at the

six cusps, P 2 → P 2. This covering is ramified along the proper transform of the Zariski curveand fits in a commutative diagram

Y −−−−→ Y

π

y π

y

P 2 −−−−→ P 2 .

Like for Y , we give preference to that complex conjugation whose real locus, Y (R), is projected

to An, call it the Mobius involution (or Mobius real structure) in Y and denote by conj = conjY.

The K3-lattice L = H2(Y ) contains a sublattice 6A2 spanned by the twelve exceptional divisorsof the resolution and the polarization class h, h2 = 2, which is represented by the pull-back ofa line in P 2. In what follows, we work with a natural ”gluing” reconstruction of L from twocomplementary sublattices: the primitive closure S ⊂ L of the sublattice spanned by 6A2 and h,and the orthogonal complement T = S⊥ = x ∈ L |xS = 0. We will consider also the sublatticeS0 = x ∈ S |xh = 0 ⊂ S and its orthogonal complement T ′ = (S0)⊥. All the sublattices, S, T ,S0, and T ′ are invariant with respect to the complex conjugation involution c = conj∗ : L → L,and we denote by L± = x ∈ L | c(x) = ±x, S± = L± ∩ S, T± = L± ∩ T , S0

± = S0 ∩ L±, and

T ′± = T ′∩L± the corresponding eigenlattices. Note that T ′

+ = T+ and S0+ = S+, since c(h) = −h

and thus h ∈ T ′− ∩ S−. It follows also that T− (respectively, S0

−) is the orthogonal complementof h in T ′

− (respectively, in S−).In particular, we obtain a relation to the ranks r± introduced in Section 2.8.

4.1.1. Lemma.

r+ =rankT+,

r− =rankT− + 1,

Proof. This follows from that H∗(Y ;Q) = H∗(Y ;Q)/(6A2 ⊗Q) = Qh+ T ⊗Q and h ∈ L−.

4.2. Deficiency in the Smith inequalities. As it follows from the Smith theory applied to thecomplex conjugation involution, the relations b∗(X(R);Z/2) 6 b∗(X ;Z/2) and b∗(X(R);Z/2) =b∗(X ;Z/2) mod 2 (here, as before, X and X(R) stand for the set of complex and real pointsrespectively) hold for any complex algebraic variety X defined over R. The variety X is called M-variety if b∗(X(R);Z/2) = b∗(X ;Z/2), and (M-d)-variety where d = 1

2 (b∗(X ;Z/2)−b∗(X(R);Z/2))otherwise (for instance, one speaks on (M-d)-curves, (M-d)-surfaces, etc.).

First of all, let us compare d(A) = 5 − ℓ(A) of a real Zariski sextic A and d(Y ) = 12 (12 −

b∗(Y (R);Z/2)) of the double covering K3-surface Y .

4.2.1. Lemma. If we choose in Y the Mobius real structure, then d(Y ) = 5− ℓ(A). Otherwise(for the non-Mobius real structure), d(Y ) = 6− ℓ(A).

Proof. Since Y (R) projects to A± as an orientation double covering with boundary glued to itselfvia deck transformation, it follows that b∗(Y (R);Z/2) = 2b∗(A±;Z/2), which is 2 + 2ℓ(A) in thecase of An (i.e., Mobius real structure), and 2ℓ(A) in the case of Ao.

Since the links of cusps are Z/2-homology spheres, the variety Y is a Z/2-homology man-ifold. Thus, its Z/2-valued intersection form 〈x, y〉, x, y ∈ H2(Y ;Z/2), is well-defined andnon-degenerate. One can also twist the intersection form via the involution c : H2(Y ;Z/2) →H2(Y ;Z/2) and define 〈x, y〉c = 〈x, c(y)〉.

We let Hc2(Y ;Z/2) = x ∈ H2(Y ;Z/2) | c(x) = x and put, similarly to the notation in Section

3.11, r2(H2(Y ;Z/2), c) = rankH2(Y ;Z/2)/Hc2(Y ;Z/2).


4.2.2. Lemma. If c : H2(Y ;Z/2) → H2(Y ;Z/2) is induced by a complex conjugation in Y suchthat Y (R) 6= ∅, then d(Y ) = r2(H2(Y ;Z/2), c).

Proof. Since H1(Y ;Z/2) = H3(Y ;Z/2) = 0 and the Smith sequence

· · · →Hr+1(Y/ conj, Y (R);Z/2) → Hr(Y/ conj, Y (R);Z/2)⊕Hr(Y (R);Z/2)trr + inr−−−−−→

→Hr(Y ;Z/2)prr−−→ Hr(Y/ conj, Y (R);Z/2) → . . .

is exact, it is sufficient to show that the image of tr2 + in2 is equal toHc2(Y ;Z/2). Such an equality

follows from the relation tr∗ pr∗ = 1 + c and, again, the exactness of the Smith sequence.

Remark. If Y (R) = ∅ then d(Y ) = r2(H2(Y ;Z/2), c) − 2. Here, one can bypass the Smiththeory and argue a bit differently. Namely, since the quotient Y/ conj is a Z/2-manifold, theimage of H2(Y/ conj;Z)/Tors by the Gysin homomorphism in H2(Y ;Z)/Tors is a lattice ofthe form L′(2) where L′ has an odd discriminant. Therefore, this lattice is primitively em-bedded in H2(Y ;Z)/Tors, which implies that r2(H2(Y ;Z), c) = rankL′. It remains to noticethat Proposition 3.11.3(2) implies r2(H2(Y ;Z), c) = r2(H2(Y ;Z/2), c), and that rankL′ =dimH2(Y/ conj;Q) = 1

2 dimH∗(Y ;Q)− 2 = 12 dimH∗(Y ;Z/2)− 2. ⊡

4.2.3. Lemma. The Gysin homomorphism ρ! : H2(Y ;Z/2) → H2(Y ;Z/2) induced by the pro-

jection ρ : Y → Y is a monomorphism and its image is T ′ ⊗Z/2 = T ′/2T ′ ⊂ L/2L = L⊗Z/2 =

H2(Y ;Z/2).The isomorphism H2(Y ;Z/2) → T ′/2T ′ provided by ρ! commutes with the involutions induced

by the complex conjugation in Y and Y and preserves the Z/2-valued intersection forms. It sendsHc

2(Y ;Z/2) to (T ′+ + T ′

−)⊗ Z/2 and induces an isomorphism

H2(Y ;Z/2)/Hc2(Y ;Z/2) → (T ′/(T ′

+ + T ′−))⊗ Z/2 = T ′/(T ′

+ + T ′−).

Proof. It follows from ρ∗ ρ! = id and absence of 2-torsion in H2(Y ) and H2(Y ).

4.2.4. Corollary. For any real Zariski sextic A and Mobius involution in Y we have r2(T+) =

r2(T′, c) = d(Y ) = d(A). If A(R) 6= ∅, and we choose a non-Mobius involution in Y , then

r2(T+) = r2(T′, c) = d(A) + 1.

Proof. From Proposition 3.11.3 it follows that r2(T′+) = r2(T

′, c) = r2(T′2, c). Since T

′+ = T+ and

r2(T′2, c) = r2(H2(Y ;Z/2), c), the remaining parts of the statement follow from Lemmas 4.2.1,

4.2.2, and 4.2.3.

4.3. Real varieties of type I. The following is a version of the so-called Arnold lemma, whichis valid for any real algebraic surface Y that is a Z/2-homology manifold and, in particular, inour case of the double plane Y → P 2 branched along Zariski sextic.

4.3.1. Lemma. The fundamental class [Y (R)] ∈ H2(Y ;Z/2) is the characteristic class of thebilinear form 〈x, y〉c, that is, for any x ∈ H2(Y ;Z/2) we have 〈x, x〉c = 〈x, [Y (R)]〉c.Proof. We pick a Z/2-cycle C representing an element x ∈ H2(Y ;Z/2), which is smooth at thenon-singular points of Y and generic with respect to Y (R), and then check that any intersectionpoint of C with Y (R) gives an odd contribution to both 〈x, c(x)〉 and 〈x, [Y (R)]〉. At the smoothpoints of Y (R) genericness means transversality and this contribution is 1. At the singular points,the intersection number can be replaced by the linking number (here, genericness means thatthe links of C, c(C), and Y (R) are disjoint). Namely, in a 3-link of each real singular point wecan fill the local link, lk(C), of C by a smooth 2-membrane, MC , lk(C) = ∂MC , transversal to


its conjugate, lk(c(C)) = ∂c(MC), and notice that MC ∩ c(MC) is a c-equivariant 1-manifoldbounding the 0-cycle (C ∩ c(MC)) ∪ (c(C) ∩MC). Finally, it remains to observe that each non-closed connected component of this 1-manifold fixed by c intersects Y (R) at one point, eachclosed connected component fixed by c intersects Y (R) at two points, while the other (non fixed)components come in pairs and are disjoint from Y (R). This implies that the linking number oflk(C) with lk(Y (R)) has the same parity as with lk(c(C)).

4.3.2. Corollary. The following conditions are equivalent.

(1) The fundamental class [Y (R)] ∈ H2(Y ;Z/2) vanishes.(2) The c-twisted intersection form is even, that is 〈x, x〉c = 0 for all x ∈ H2(Y ;Z/2).

A real algebraic surface (Y, c) is said to be of type I if the conditions of Corollary 4.3.2 aresatisfied, otherwise, we say that it is of type II.

4.3.3. Lemma. The following conditions are equivalent.

(1) The pair (Y, c) is of type I;(2) The pair (T ′, c|T ′) is of type I;(3) The discriminant form in discr2(T

′+) = discr2(T+) is even, or in the other words, δ2(T+) =

0.

Proof. Since the Gysin homomorphism ρ! : H2(Y ;Z/2) → H2(Y ;Z/2) commutes with conj, pre-serves the Z/2-intersection form and, by Lemma 4.2.3, establishes an isomorphism betweenH2(Y ;Z/2) and T ′/2T ′, the equivalence between (1) and (2) follows from Corollary 4.3.2. Theequivalence between (2) and (3) follows from Proposition 3.11.3.

4.3.4. Lemma. A Zariski sextic A is of type I if and only if Y is of type I with respect to theMobius real structure.

Proof. Due to Corollary 4.3.2 and exactness of the Smith sequence written for the deck trans-formation of the double covering Y → P 2, it is sufficient to check that [An] ∈ H2(P

2, A;Z/2)is zero if A is of type I (note that it is non zero, otherwise). But if A is of type I, one canapply Rokhlin’s trick: consider instead [An] + [RP 2] = [Ao], lift the latter up to an integerhomology cycle R = [Ao] + [A±] ∈ H2(P

2) (here, A± denote the two components of Ar A(R))and check that R = 1

2 (R − conj∗R) =12 [A] ∈ H2(P

2), while 12 [A] reduced modulo 2 is equal to

[RP 2] ∈ H2(P2;Z/2).

4.3.5. Proposition. A Zariski sextic A is of type I if and only if δ2(T+(Y )) = 0, where Y isendowed with the Mobius real structure.

Proof. It is a straightforward consequence of Lemmas 4.3.4 and 4.3.3.

4.4. Resolution decoration of H2(Y ). Given a K3-lattice L = 3U+ 2E8, we say that (∆, h)is a decoration of L and that L is (∆, h)-decorated, if ∆ ⊂ L is a set of twelve elements calledthe exceptional classes forming a basis of 6A2 (i.e., ∆ = e′1, e′′1 , . . . , e′6, e′′6, (e′i)2 = (e′′i )

2 = −2,e′ie

′′i = 1, and e′ie

′j = e′ie

′′j = e′′i e

′′j = 0 for 1 6 i 6= j 6 6) and h ∈ L is an element orthogonal to

∆ such that h2 = 2.In the case of the K3-surface Y obtained by resolution of singularities of the double plane

Y → P 2 ramified along a Zariski sextic 4p3+27q2 = 0, such kind of decoration appears naturally

in L = H2(Y ) ∼= H2(Y ): the exceptional curves lying above the six A2-singularities of Y provide

the exceptional classes e′i, e′′i ∈ ∆, and h ∈ L is the polarization class of the projection Y → Y →

P 2 (it can be geometrically thought of as the pull-back of a generic line in P 2). We call thisdecoration a resolution decoration.

This resolution, and hence the decoration, can be enriched by a consideration of the pullbackof the conic p = 0. Indeed, the resolution of the cusps together with the pullback of the conic can


be obtained in two steps: first, blowing up of P 2 at the six cups, and, second, taking the doublecovering ramified along the proper transform of the Zariski sextic. As a result, if the conic isnonsingular, we get as its pullback the two (−2)-curves that are proper transforms of the conicand the twelve exceptional curves E′

i, E′′i . Let us denote these two (−2)-curves by Q′, Q′′. Under

appropriate ordering, the Dynkin-Coxeter graph of this collection of curves looks as on Figure3a).

' ''

' ''

' ''

' ''

' ''

' ''

1

2

3

4

5

6

1

2

3

4

5

6' ''

' ''' ''

' ''

' ''

' ''

1

2

3

4

5

6

1

2

3

4

5

6

a) b)

Q Q

L

L

L

L'

'

''

''1 1

2 2

' ''

E

E

E

E

E

EE

E

E

E

E

E

E

E

E

E

E

EE

E

E

E

E

E

Figure 3.

If the conic is singular, it splits in a union of two distinct lines, each containing three of thesix cusps. The pullback of each line splits into two (−2)-curves. Let us denote these two pairs byL′1, L

′′1 and L′

2, L′′2 . Under an appropriate ordering, L′

1 intersects L′2 and L′′

1 intersects L′′2 . We

fix such an ordering and put Q′ = L′1 + L′

2, Q′′ = L′′

1 + L′′2 . The Dynkin-Coxeter graph of the

collection L′1, L

′2, L

′′1 , L

′′2 , E

′1, . . . E

′′6 is as on Figure 3b).

4.4.1. Lemma. For any Zariski sextic, the following relations hold in L = H2(Y ):

[Q′] = h−∑ e′′i + 2e′i

3, [Q′′] = h−

∑ e′i + 2e′′i3

.

Proof. Since the both sides in each of the identities to be proved belong to sublattice S, the resultfollows just from a straightforward verification that the both sides have the same intersectionindices with h, e′1, e

′′1 , . . . , e

′6, e

′′6 generating S, which in turn follows from the incidence relations

shown on the above Dynkin-Coxeter graphs.

4.5. The Galois S3-coverings. One of the consequences of Lemma 4.4.1 is that 6A2 is not aprimitive sublattice of L. A more conceptual explanation of this non-primitiveness consists inappealing to the Galois covering Z → P 2 with the Galois group S3 (symmetric group on a set ofthree elements) induced by the central projection πX : X → P 2, that is the Galois covering withthe Galois group S3 branched along the Zariski curve A of πX : X → P 2 and whose unramifiedpart is formed by the fibers Zs, s ∈ P 2 \ A, consisting of the six orderings of the 3-elementssets π−1

X (s) (in algebraic terms, it is the ramified Galois covering whose Galois group is themonodromy group of πX). This covering has the following remarkable properties:

(1) Z is a nonsingular K3-surface with a holomorphic S3-action;(2) the subgroup C3 ⊂ S3 formed by even permutations acts on Z symplectically (i.e., trivially

on the holomorphic differential 2-forms);(3) the quotient Z/C3 is canonically identified with Y ;(4) the C3-action has precisely six fixed points and they are the pullback of the six cusps of

Y .


In particular, we obtain the following diagram of projections

Z −−−−→ Yy

yπY

XπX−−−−→ P 2.

Just an existence of such a nontrivial cyclic order three covering Z → Y implies that 6A2 isnot primitive in L. Namely, we obtain the following well-known property.

4.5.1. Lemma. The primitive closure S0 of 6A2 in L is an extension of index 3.

Proof. Since the discriminant order | discr 6A2| = 36 is a power of 3, the index [S0 : 6A2] is alsoa power of 3 and, therefore, coincides with the order of H1(Y rSing Y ;Z/3). The latter group isnontrivial, since a connected C3-covering of Y r Sing Y is provided by Z → Y . It is isomorphicto Z/3, since Z is smooth at the six points z1, . . . , z6 that form the pull back of Sing Y and bythis reason Z r z1, . . . , z6 does not admit any further nontrivial C3-covering.

4.5.2. Proposition. For any resolution decoration of L = H2(Y ),

(1) the order in each pair of A2-generators e′i, e

′′i ∈ ∆ can be chosen in such a way that the

element σ = (e′1 − e′′1) + · · ·+ (e′6 − e′′6) is divisible by 3 in L;(2) the primitive closure S0 is spanned by ∆ ∪ σ

3 ;(3) the sublattice S spanned by ∆ ∪ σ

3 , h in L is primitive and splits into a direct sum

〈2〉+ S0, where 〈2〉 is generated by h.

Proof. Statements (1) and (2) are immediate consequences of Lemmas 4.4.1 and 4.5.1. Statement(3) follows from absence of 2-torsion in the discriminant of S0.

4.5.3. Corollary. (1) discr(S0) = 〈23 〉+ 3〈− 23 〉.

(2) S has the following direct sum decompositions: S = 〈2〉+S0 = 〈2〉+E6+3A2 = U+A5+3A2.

Proof. By Proposition 4.5.2(2) and Lemma 3.8.1, we have discrS0 = [σ3 ]⊥/[σ3 ], where [σ3 ] ∈

discr 6A2 = 6〈− 23 〉 is the diagonal element. It follows now from Lemma 3.7.2 that discrS0 =

p〈23 〉+ q〈− 23 〉, p+ q = 4, where 2(p− q) = Br(S0) = Br(6A2) = 4 mod 8, which gives (1).

Part (2) follows from Theorem 3.12.1, since discr(S0) = 〈− 23 〉+3〈23 〉, discrE6 = 〈23 〉, discrA2 =

〈− 23 〉, and discrA5 = 〈− 5

6 〉 = 〈12 〉+ 〈23 〉.

4.5.4. Proposition. The Galois covering Z → P 2 inherits a real structure, cZ : Z → Z, fromthe real structure of the cubic surface X. This real structure, cZ , commutes with the Galois

action of C3 ⊂ S3 on Z and descends to real structures cY : Y → Y , cY

: Y → Y (as Y is

identified with Z/C3) commuting with the deck transformation of Y → P 2 and the resolution of

singularities Y → Y ; the real parts Y (R) and Y (R) are projected to A− ⊂ P 2R.

Conversely, if Y (R) is nonempty and a real structure c : Y → Y commutes with the involution

Y → Y induced by the deck transformation Y → Y , then c lifts to a real structure cZ : Z → Z.Moreover:

(1) if c∗(σ) = σ, then the above cZ commutes with the Galois action of C3;(2) if c∗(σ) = −σ, then cZ together with the C3-action form an action of a semi-direct

product, C3 ⋊ Z/2 = S3.

Proof. The direct statements is a straightforward consequence of the construction of Galois cov-erings induced by a given projection. The assumption Y (R) 6= ∅ in the converse statement is onlyfor ensuring that the lift of c is an involution. Finally, the commutator relation cZ θ = θo(c) cZ ,where o(c) is defined by (cZ)∗(σ) = o(c)σ, follows from the Poincare-Lefschetz duality (that trans-forms 1

3σ into a characteristic element of the Galois covering over Y rSing Y , cf., proof of Lemma4.5.1) and the construction of cyclic coverings.


4.6. Abstract K3-lattice conical (∆, h)-decorations. We say that a (∆, h)-decoration of aK3-lattice L is conical if it satisfies the properties (1)–(2) (and thus, also (3) ) of Proposition4.5.2. According to this Proposition, each resolution decoration is conical.

The properties (1) and (2) in Proposition 4.5.2 imply that, for a given conical (∆, h)-decorationof L, the choice of σ is unique up to sign. We call σ the ∆-master element. Thus, the choiceof a ∆-master element determines the order (e′i, e

′′i ) of generators in each of the A2-components

of ∆, but not the order of the indices 1 6 i 6 6. The choice of the opposite ∆-master elementalternates the order of A2-generators.

Given a conically (∆, h)-decorated K3-lattice L, we denote by T and T ′ (in accordance withSection 4.1) the orthogonal complements of sublattices S and S0, respectively.

4.6.1. Lemma. The following properties hold for any K3-lattice with a conical (∆, h)-decoration.

(1) T ′ is an extension of T+〈2〉 of index 2. Namely, T ′ is obtained by adding to T an element12 (h+ x) where x ∈ T is a primitive element and x.T ⊂ 2Z.

(2) There are following isomorphisms

T = U+ U(3) + 2A2 + A1 = 〈6〉+ U+ 3A2 = A2(−1) + 3A2 + A1,

T ′ = 2U+ U(3) + 2A2.

Proof. Since discr(S0) and discr(T ′) are isomorphic as groups, the first statement follows from theabsence of 2-torsion in discr(S0). The second statement follows from Theorem 3.12.1, discr(T ′) =− discr(S0) = 〈23 〉 + 3〈− 2

3 〉, discrU(3) = 〈23 〉 + 〈− 23 〉, discrA2 = 〈− 2

3 〉, discr(T ) = − discr(S) =

〈− 12 〉+ discr(T ′), discr〈6〉 = 〈16 〉 = 〈− 1

2 〉+ 〈23 〉, and Lemma 3.7.2.

By an isomorphism between a pair of K3-lattices, L1 and L2, decorated with (∆1, h1) and(∆2, h2) respectively, we mean a lattice isometry f : L1 → L2 such that f(∆1) = ∆2 and f(h1) =h2.

We denote by Aut(L,∆, h) the group of automorphisms of a (∆, h)-decorated K3-lattice L.We let also Aut(L,∆, h) = f | − f ∈ Aut(L,∆, h).4.6.2. Lemma. If a (∆, h)-decoration of a K3-lattice L is conical, then for a ∆-master elementσ and any f ∈ Aut(L,∆, h) we have f(σ) = ±σ.Proof. Straightforward consequence of the definitions (properties (1) and (2) in Proposition4.5.2).

For any f from Aut(L,∆, h) or Aut(L,∆, h) we define o(f) ∈ +,− by imposing the relationf(σ) = o(f)σ.

4.6.3. Corollary. Let c : L → L be induced by the Mobius involution in Y . Then c ∈Aut(L,∆, h) and o(c) = o(A), where A is the Zariski sextic of Y .

Proof. Due to Proposition 4.5.4, if f ′ : L → L is induced by cZ , then f ′ ∈ Aut(L,∆, h) ando(f ′) = +. Thus, there remains to notice that o(A) = + if and only if p : X(R) → P 2(R) is

three-fold over the nonorientable half of P 2(R), and that the deck transformation of Y → P 2

reverses the master element σ. The latter follows from Lemma 4.4.1, since the deck transformationpermutes Q′, Q′′.

4.6.4. Proposition. If Li, i = 1, 2 are conical (∆i, hi)-decorated K3-lattices, then they areisomorphic as decorated K3-lattices.

Proof. Using an isomorphism for T in Lemma 4.6.1, we obtain a required isomorphism of K3-lattices applying Lemma 3.9.4.


4.7. S0-eigenlattices. As above, let us consider a real Zariski sextic A, the Mobius involution

conj on Y , and the induced involution c : L→ L. The latter obviously preserves each of the sub-lattices S, T , S0, T ′ invariant. Let us mark with indices ± (e.g., S±, T±, etc.), the corresponding±1-eigenlattices.

If in a lattice M its element v ∈ M is not divisible by d > 1, but vx is divisible by d for allx ∈M , and v2 is divisible by d2, then by adding element v

dto L we obtain its extension (still an

integral lattice) of index d, which we denote by [M ] vd. In this notation, S0 = [6A2]σ

3.

4.7.1. Proposition. The eigenlattices S0± are determined up to isomorphism by the number νi

of imaginary pairs of cusps of A and the sign o(A) ∈ +,−, as is indicated in the Table 2.

Table 2

The case of o = −νi S0

+ S0−

— ——— ———0 0 [6A2]σ

3

1 A2(2) [4A2 + A2(2)]σ3

2 2A2(2) [2A2 + 2A2(2)]σ3

3 3A2(2) [3A2(2)]σ3= E6(2)

The case of o = +

νi S0+ S0

−

— ——— ———0 [6〈−6〉]σ

36A1

1 [4〈−6〉+ A2(2)]σ3

4A1 + A2(2)2 [2〈−6〉+ 2A2(2)]σ

32A1 + 2A2(2)

3 [3A2(2)]σ3= E6(2) 3A2(2)

Proof. As follows from Corollary 4.6.3, the master element σ defining the extension S0 = [6A2]σ3is preserved by c if o = +, and reversed otherwise. If o = −, then each real cusp gives A2 ⊂ S0

−.If o = +, then at each real cusp we have c(e′i) = −e′′i , so that e′i + e′′i ∈ S0

− and e′i − e′′i ∈ S0+. A

pair of imaginary cusps gives a copy of A2(2) in S0+ and another copy in S0

−.

4.7.2. Corollary. For any real Zariski sextic and the associated eigenlattices S0±, we have

discr3 S0+ = p〈−2

3〉+ q〈2

3〉, discr3 S

0− = (1− p)〈−2

3〉+ (3 − q)〈2

3〉

where p = 1, 0 6 q 6 3 if o(c) = + and p = 0, 0 6 q 6 3 if o(c) = −.Under such a presentation, the number νi of imaginary pairs of cusps is q if p = 1, and 3− q

if p = 0.

A pair of lattices isomorphic to one of the eight pairs (S0+, S

0−) in Table 2 will be called an

S-pair. Each component, S0±, of such a pair will be called an S-half.

4.8. Geometric involutions. We say that a lattice hyperbolic if its positive inertia index equals1 (with the usual abuse of terminology if the negative inertia index is 0). We say that an involutionc : L → L in a conical (∆, h)-decorated K3-lattice L is geometric if the following conditions aresatisfied:

(1) c ∈ Aut(L,∆, h);(2) the eigenlattices T±(c) are hyperbolic.

4.8.1. Lemma. For any geometric involution c : L→ L of a conical (∆, h)-decorated K3-latticeL, the pair of eigenlattices T± = T±(c) has the following properties:

(1) r(T+) + r(T−) = 9;(2) T± have no other discriminant factors than 2 and 3, and the latter ones are both elemen-

tary;(3) |r2(T+)− r2(T−)| = 1,(4) the eigenlattice whose rank r2 is greater has δ2 = 1;(5) discr3 T+ + discr3 T− = 〈23 〉+ 3〈− 2

3 〉.


Proof. Statement (1) follows from r(S) + r(T+) + r(T−) = r(L). Statement (2) follows fromLemma 3.10.1 applied to h ∈ T ′ and Proposition 3.11.1. If T ′

− = Zh + T− then discr2(T+) =

−〈12 〉 + discr2(T−) and r2(T+) = r2(T−) + 1, otherwise, T ′− is obtained by a nontrivial gluing

Zh+φ T− and then r2(T+) = r2(T−)− 1, so that in both cases we get (3) and (4). Statement (5)follows from discr(S0) = 〈− 2

3 〉+ 3〈23 〉 (see Corollary 4.5.3) and Proposition 3.11.1.

Let us call a geometric involution ascending if r2(T+) < r2(T−) and descending otherwise.

Similarly, a complex conjugation on Y is called ascending involution or ascending real structureif the induced by it geometric involution is ascending; otherwise, we call it descending involutionor descending real structure.

4.8.2. Lemma. Assume that A is a Zariski sextic, and c is the involution in L = H2(Y ) induced

by one of the two lifting of the complex conjugation from P 2 to Y . Then:

(1) c is geometric;(2) if c is induced by the Mobius real structure and A(R) 6= ∅, then c is ascending;(3) if c is induced by the non-Mobius real structure and A(R) 6= ∅, then c is descending.

Proof. The involution c is geometric, since each lift of the complex conjugation from P 2 to Ysends the exceptional divisors and the polarization to the exceptional divisors and the polariza-tion, reversing their orientation, and as any anti-holomorphic involution, permutes the Hodgesummands, H2,0 and H0,2. Since r2(T+) = r2(T

′+) = r2(T

′, c), Corollary 4.2.4 implies (2) and(3).

Let C(L,∆, h) denote the set of geometric involutions, and let C<(L,∆, h), C>(L,∆, h) de-note the set of the ascending and descending ones respectively. The group Aut(L,∆, h) actson C(L,∆, h) preserving the subsets C<(L,∆, h) and C>(L,∆, h) invariant. Let C[L,∆, h],C<[L,∆, h], and C>[L,∆, h] denote the orbit spaces (i.e., the set of conjugacy classes) of geo-metric involutions, ascending ones, and descending ones respectively. We say that two geometricinvolutions have the same homological type, if they represent the same element in C[L,∆, h].

4.9. T-pairs and T-halves. We say that a pair of even hyperbolic lattices (T1, T2) form aT-pair if they satisfy the five properties stated in Lemma 4.8.1 for (T+, T−). A T-pair will becalled ascending (descending) if r2(T1) < r2(T2) (respectively r2(T1) > r2(T2)).

4.9.1. Lemma. For any ascending T-pair (T1, T2) or descending T-pair (T2, T1)

r2(T1) 6 min(r(T1), 8− r(T1)),

r2(T2) 6 min(r(T2), 10− r(T2)).

In particular, r2(T1) 6 4, and r2(T2) 6 5, where r2(T2) = 5 implies that r(T1) = r2(T1) = 4 andr(T2) = 5.

Proof. It follows from properties (1) and (3) combined with the bounds r2(Ti) 6 r(Ti), seeProposition 3.4.1.

A lattice is called a T-half if it is a component, T1 or T2, of some T-pair. The following is astraightforward consequence of Lemma 4.8.1.

4.9.2. Proposition. If M is a T -half, then:

(1) M is hyperbolic of rank 1 6 r 6 8;(2) M has no discriminant p-factors different from p = 2, 3, and the latter ones (if exist) are

elementary;(3) M has either r2 6 4, or r2 = r = 5, and in the latter case, it can be only the second

component of an ascending T-pair;(4) discr3M = p〈23 〉+ q〈− 2

3 〉, where 0 6 p 6 1 and 0 6 q 6 3.


4.9.3. Proposition. For any c ∈ C(L,∆, h), the S0-eigenlattices (S0+, S

0−), the T -eigenlattices

(T+, T−), and T′− must satisfy the following relations.

(1) The pair (S0+, S

0−) is an S-pair.

(2) T± are T-halves and either (T+, T−) or (T+, T−) is an ascending T-pair.(3) discr3 T± is anti-isomorphic to discr3 S

0±, and in particular r3(T±) = r3(S

0±).

(4) discr2 T+ is anti-isomorphic to discr2 T′−, and in particular |r2(T+)− r2(T−)| = 1.

Proof. Any c ∈ C(L,∆, h) by definition preserves sublattice 6A2 ⊂ S0 invariant. Moreover, anautomorphism of 6A2 must permute its A2-components, and then, the arguments similar to thatof Proposition 4.7.1 can be applied to obtain (1). (2) follows immediately from definitions. Sincediscr(L±) for the ±-eigenlattices L± are elementary 2-groups as it follows from Proposition 3.11.1,gluing of L± from T±, and S± involves an anti-isomorphism of their discr3-components accordingto Proposition 3.9.3, which implies (3). Since discrT ′ is a 3-group, the discr2-components of T+and T ′

− are anti-isomorphic, again by Proposition 3.9.3, which implies (4).

Property (3) in Proposition 4.9.3 shows that the pair (p, q) describing discr3(T+) (or equiva-lently, discr3(T−)) determines and is determined by the pair r3(T+) = p+ q and o(c). Namely, itshows that Corollary 4.7.2 can be restated as follows.

4.9.4. Corollary. The ranks 0 6 p 6 1 and 0 6 q 6 3 in the decompositions discr3 T+ =p〈23 〉+ q〈− 2

3 〉, discr3 T− = (1− p)〈23 〉+(3− q)〈− 23 〉 determine o(c) and the number νi as follows:

(1) if p = 1, then o(c) = + and νi = q;(2) if p = 0, then o(c) = − and νi = 3− q.

4.9.5. Lemma. For any T-pair (T1, T2), Br2(T1) + Br2(T2) = −1.

Proof. Applying Corollary 3.7.4 to Ti, i = 1, 2, we get Br2(Ti) + Br3(Ti) = (2 − r(Ti)), whereBr3(Ti) = 2(pi − qi) if discr3 Ti = pi〈23 〉 + qi〈− 2

3 〉. Using r(T1) + r(T2) = 9 and discr3 T1 +

discr3 T2 = 〈23 〉+ 3〈− 23 〉 we conclude that Br2(T1) + Br2(T2) = (4− 9)− 2(1− 3) = −1.

4.9.6. Lemma. Assume that (T1, T2) is an ascending T-pair. Then K1 = discr2 T1 is anti-isomorphic to a subgroup K2 ⊂ discr2 T2 if and only if K2 is the orthogonal complement v⊥ ofan element v ∈ discr2 T2 satisfying the following conditions:

(1) qT2(v) = − 12 ∈ Q/2Z;

(2) v is a Wu element if and only if δ2(T1) = 0.

Proof. If K2 is anti-isomorphic to K1, then K2 is a non-degenerate subgroup of discr2 T2 ofcorank r2(T2) − r2(T1) = 1 and, therefore, K2 = v⊥ for some v ∈ discr2 T2. Furthermore,Br2(T1) = Br(K1) = −Br(K2), and additivity of Br implies Br2(T2) = Br(K2) + Br〈v〉. andBr〈v〉 = 2qT2(v) mod 4 by Lemma 3.6.4(3). Thus, Br2(T1) + Br2(T2) = 2qT2(v) mod 4, whichequals−1 mod 4 by Lemma 4.9.5 and thus gives condition (1). Lemma 3.6.4(2) implies condition(2).

Conversely, if v ∈ discr2 T2 satisfies condition (2), then δ2(K2) = δ2(K1). If v satisfies condition(1), then Br〈v〉 = −1 (see Lemma 3.6.4(4)), and thus Br(K2) = Br2(T2) + 1. Applying Lemma4.9.5 we get Br(K1) = Br2(T1) = −1 − Br2(T2) = −Br(K2) and by means of Theorem 3.6.3conclude that K1 is anti-isomorphic to K2.

4.10. T-halves and pairs of Mobius involutions. Given a Zariski sextic A, let c1 denote

the Mobius involution in L = H2(Y ), and c2 be the non-Mobius one.

4.10.1. Proposition. T±(c2) = T∓(c1) for any Zariski sextic A. Moreover, in the case A(R) =∅,

(1) T+(c1) = T−(c2) have r = r2 = 5 and δ2 = 1;(2) T−(c1) = T+(c2) have r = r2 = 4 and δ2 = 0.


Proof. Note that c2 c1 = c1 c2 is induced by the deck transformation of the covering Y → P 2,

and the action of the deck transformation in H2(Y ) is equal to the product of pairwise commutingmaps −ρh and ρe′

i+e′′

i, i = 1, . . . , 6, where ρv stands for the reflection x 7→ x − 2xv

v2 v. Therefore,

T±(c2) = T∓(c1).

In the case of A(R) = ∅, Lemma 2.8.1 gives values r+ = r− = d(A) = 5 for the involution c1,and applying Lemma 4.1.1 we obtain the values of r(T±(c1)) = r(T±(c2)) indicated in (1)–(2).

Corollary 4.2.4 implies that r2(T+(c1)) = 5. The parity δ2 = 0 for T+(c2), and thus for T−(c1),follows from Corollary 4.3.2. The imparity δ2 = 1 for T+(c1), and thus for T−(c2), follows fromProposition 4.3.5, since A(R) = ∅ implies that A has type II. Finally, the values of r2(T±(c1)),and thus of r2(T±(c2)), indicated in (1)–(2) follow from Lemmas 4.2.1 and Corollary 4.2.4.

5. Arithmetics of geometric involutions

5.1. Automorphisms of 3-elementary inner product groups of small rank. Here, weanalyze the automorphisms of discr(S0) ∼= 〈− 2

3 〉+ 3〈23 〉 ∼= 〈23 〉+ 3〈− 23 〉 (see Corollary 4.5.3) and

of its non-degenerate subgroups.

Note that each permutation of coordinates followed by an alternation of signs of some com-ponents, (x1, . . . , xn) 7→ (±xσ(1), . . . ,±xσ(n)), provides an automorphism of n〈23 〉, as well as of

n〈− 23 〉. These ”coordinatewise” automorphisms form a group that we denote by Autcomp(n〈± 2

3 〉).It is isomorphic to the group of symmetries of an n-cube and fits canonically in an exact sequence1 → (Z/2)n → Autcomp(n〈± 2

3 〉) → Sn → 1.

5.1.1. Lemma. If 1 6 n 6 3, then Aut(n〈± 23 〉) = Autcomp(n〈± 2

3 〉).

Proof. Note that for q = n〈± 23 〉 the expression q(x1, . . . , xn) = ± 2

3 (x21 + · · · + x2n) (mod 2Z)

depends only on the number of non-zero coordinates modulo 3. Thus, for n 6 3 the elements ofthe direct summands in n〈± 2

3 〉 are distinguished from all the other elements if n 6 3. Hence,

any automorphism in Aut(n〈± 23 〉) with n 6 3 is coordinatewise.

Now, consider an enhanced group (G, q) = 6〈− 23 〉 and denote by δ =

∑6i=1 ai the diagonal

element, that is the sum of the generators ai of all the 〈− 23 〉-components of G. Note that δ is

isotropic, that is δ2 = 0, and thus δ ∈ Gδ = x ∈ G |xδ = 0.We put b

i1...ipj1...jq

= (ai1 + . . . aip)− (aj1 + . . . ajq ) ∈ G, and in the case δbi1...ipj1...jq

= − 23 (p− q) = 0,

we denote by [bi1...ipj1...jq

] the coset in Gδ/(δ). Since δ is isotropic, the quotient group Gδ/(δ) inherits

from G the inner product and quadratic form.

5.1.2. Lemma. The non-trivial elements of the group Gδ/(δ) = 〈− 23 〉 + 3〈23 〉 break up into 3

sets:

(1) 30 elements [bij ], 1 6 i, j 6 6, i 6= j, of square 23 ;

(2) 30 elements [bijkl] of square − 23 , where 1 6 i, j, k, l 6 5 are distinct;

(3) 20 isotropic (i.e., of square 0) elements [bijk], 1 6 i < j < k 6 6.

Proof. Straightforward.

5.1.3. Lemma. The elements b1234,b12,b

34,b

56 split the enhanced group Gδ/(δ) in an orthogonal

direct sum 〈− 23 〉+ 3〈23 〉.

Proof. Straightforward.


5.2. Reduction homomorphism. Keeping the notation of the previous Subsection, we con-sider the automorphism group AutG and study its subgroups:

Aut(G, δ) =f ∈ Aut(G) | f(δ) = ±δ,Autcomp(G, δ) =f ∈ Autcomp(G) | f(δ) = ±δ,Aut+comp(G, δ) =f ∈ Autcomp(G, δ) | f(δ) = δ.

The induced homomorphism Aut(G, δ) → Aut(Gδ/(δ)) = Aut(〈− 23 〉+3〈23 〉) will be called the

reduction homomorphism.

5.2.1. Proposition.

(1) Aut+comp(G, δ) = S6 and Autcomp(G, δ) = S6 × Z/2.(2) The reduction homomorphism restricted to Autcomp(G, δ) is an isomorphism S6 ×Z/2 =

Autcomp(G, δ) → Aut(Gδ/(δ)) = Aut(〈− 23 〉+ 3〈23 〉).

5.2.2. Lemma. Autcomp(G, δ) acts effectively and transitively on the set of 23 -elements [bij] ∈

Gδ/(δ).

Proof. It follows from the description of 23 -elements in Lemma 5.1.2.

Proof of Proposition 5.2.1. The claim (1) is straightforward, since δ is preserved by a coordi-natewise automorphism if and only if it is defined by a simple permutation (without any signreversion).

Lemma 5.2.2 implies that the reduction homomorphism is injective on Autcomp(G, δ). Toshow that it is isomorphic, it is sufficient to check that the order of the group Aut(〈− 2

3 〉+ 3〈23 〉)coincides with that of Autcomp(G, δ).

The order of Autcomp(G, δ) is equal to |S6×Z/2| = 2 · 6!. On the other hand, by Lemma 5.2.2the group Aut(〈− 2

3 〉+3〈23 〉) acts transitively on the 30 elements [bij ]. The stabilizer of an element

[bij ] is isomorphic to Aut(3〈− 23 〉), since the orthogonal complement of [bij ] in G

δ/(δ) = 〈23 〉+3〈− 23 〉

is isomorphic to 3〈− 23 〉 (see Lemma 3.7.2). This implies that the order of Aut(〈− 2

3 〉 + 3〈23 〉) is

30|Aut(3〈23 〉)|, where by Lemma 5.1.1 |Aut(3〈23 〉)| = 8|S3| = 48.

Remark. In fact, it is not difficult to prove that Aut(G, δ) = Autcomp(G, δ), and, therefore,Proposition 5.2.1(2) means that the reduction homomorphism is an isomorphism. ⊡

Now, assume that cG ∈ S6 ×Z/2 = Autcomp(G, δ) is an involution acting on G. It induces aninvolution cδG in 〈− 2

3 〉 + 3〈23 〉 = Gδ/(δ). Let Autcomp(G, δ, cG) = f ∈ Autcomp(G, δ) | f cG =

cG f and Aut(〈− 23 〉+ 3〈23 〉, cδG) = f ∈ Aut(〈− 2

3 〉+ 3〈23 〉) | f cδG = cδG f. As an immediatecorollary, we obtain the following equivariant version of Proposition 5.2.1(2).

5.2.3. Corollary. The isomorphism in Proposition 5.2.1(2) restricts to an isomorphism

Autcomp(G, δ, cG) → Aut(〈−2

3〉+ 3〈2

3〉, cδG).

5.3. Equivariant epistability of S0. The proof of the following Lemma is straightforward.

5.3.1. Lemma. There is a canonical isomorphism discr(S0) = Gδ/(δ).

Given a subset D ⊂ L in a lattice L, we let Aut(L,D) = f ∈ AutL | f(D) = D. We say thatL is D-relatively epistable if the induced homomorphism Aut(L,D) → Aut(discrL) is surjective.


5.3.2. Proposition. The induced projection Aut(S0,∆) → Aut(discrS0) is an isomorphism.In particular, S0 is ∆-relatively epistable.

Proof. Each automorphism f ∈ Aut(S0,∆) preserves the sublattice 6A2 ⊂ S0, permutes itsA2-components, and preserves or reverses the ∆-master element σ ∈ 6A2 that is responsible forthe extension of 6A2 to S0. Thus, the induced automorphism in G = discr(6A2) preserves orreverses δ. This yields a homomorphism Aut(S0,∆) → Autcomp(G, δ) = S6 × Z/2, which is anisomorphism. Now, it is left to apply Lemma 5.3.1 and Proposition 5.2.1(2).

Let Aut(S0,∆) = f ∈ Aut(S0) | − f ∈ Aut(S0,∆).5.3.3. Corollary. The projection Aut(S0,∆) → Aut(discrS0) is bijective. Thus, any involu-tion in discrS0 can be lifted to an involution of S0 which sends ∆ to −∆.

Proof. Given φ ∈ Aut(discrS0), let f ∈ Aut(S0,∆) be a lifting of −ψ existing by Proposition5.3.2. Then −f ∈ Aut(S0,∆) is a lifting of φ that we need.

Now, consider an involution c ∈ C(L,∆, h) and the induced involutions, cG on G and cδG ondiscrS0 = Gδ/(δ). Let Aut(S0,∆, c) = f ∈ Aut(S0,∆) | fc = cf, and Aut(discrS0, cδG) =φ ∈ Aut(discrS0) |φcδG = cδGφ.5.3.4. Corollary. A restriction of the isomorphism of Proposition 5.3.2 yields an isomorphismAut(S0,∆, c) ∼= Aut(discrS0, cδG).

Proof. By definition, −c belongs to Aut(S0,∆) and Aut(S0,∆, c) coincides with the centralizerof −c, whereas Aut(discrS0, cδG) is the centralizer of its image −cδG under the isomorphismAut(S0,∆) → Aut(discrS0).

The property of S0 indicated in Corollary 5.3.4 will be referred to as c-equivariant ∆-relativeepistability of S0. In a bit more general setting, the definition looks as follows. Given a lattice Lwith a subset D ⊂ L and an involution c : L→ L inducing an involution cdiscr in discrL, we saythat L is c-equivariant D-relative epistable if the induced homomorphism from Aut(L,D, c) =f ∈ Aut(L,D) | f c = c f to Aut(discrL, cdiscr ) = φ ∈ Aut(discrL) |φ cdiscr = cdiscr fis surjective.

5.4. Gluing of involutions. If two lattices L1, L2 are glued along different anti-isomorphisms,φ and φ′ between the same subgroups Ki ⊂ discrLi, i = 1, 2, Lemma 3.9.4 gives the followingcriterion of existence of an isomorphism f : L→ L′ with f(Li) = Li for i = 1, 2.

5.4.1. Lemma. Assume that L = L1 +φ L2 and L′ = L1 +φ′ L2, where φ and φ′ are anti-isomorphisms K1 → K2 between the same subgroups Ki ⊂ discrLi, i = 1, 2. Then, a pair ofautomorphisms fi : Li → Li, i = 1, 2 can be extended to an isomorphism f : L → L′ if and onlyif fi are (φ, φ′)-compatible, that is if and only if the following two conditions are satisfied:

(1) for each i = 1, 2, the automorphism (fi)∗ induced by fi on discrLi preserves Ki;(2) (f2)∗|K2 φ = φ′ (f1)∗|K1 .

If we let L′ = L using one of such isomorphisms for identification, then all the others can becharacterized as follows.

5.4.2. Corollary. Automorphisms f : L → L of L = L1 +φ L2, φ : K1 → K2, such thatf(Li) = Li, i = 1, 2, are in one-to-one correspondence with the pairs (f1, f2) of (φ, φ)-compatibleautomorphisms of L1 and L2 (here, the compatibility means that the induced homomorphisms(f1)∗ : discrL1 → discrL1, (f2)∗ : discrL2 → discrL2 preserve K1,K2 and, being restricted toK1,K2, commute with φ).

Next, we give a criterion for existence of an extension of an automorphism f1 : L1 → L1 fromL1 to L = L1 +φ L2.


5.4.3. Proposition. Let L = L1 +φ L2, φ : K1 → K2, be a gluing such that K2 ⊂ discrL2 isa direct summand, and lattice L2 is epistable. Assume that f1 : L1 → L1 is an automorphism ofL1 and (f1)∗(K1) = K1.

(1) Then f1 can be extended to an automorphism f : L→ L.(2) Furthermore, if L′ = L1+φ′L2, φ

′ : K1 → K2, then f1 can be extended to an isomorphismf : L→ L′.

Proof. Here (1) is a special case of (2), so we shall just prove the latter. Let us define ψ2|K2 =φ′(f1)∗|K1φ−1 and extend it to an automorphism ψ2 : discrL2 → discrL2 using the assumptionthat K2 splits off as a direct summand. Due to epistability of L2, we can find an automorphismf2 : L2 → L2 such that ψ2 = (f2)∗. Now, the assumptions of Lemma 5.4.1 are satisfied and weobtain f : L→ L′ by gluing f1 and f2.

Let Li be equipped with involutions ci and cdiscri denote the induced involutions on discrLi.

An anti-isomorphism φ : K1 → K2 between ci-invariant subgroups Ki ⊂ discrLi is said to beequivariant if it commutes with cdiscri |Ki

. The following lemma is also an immediate consequenceof Lemma 3.9.4

5.4.4. Lemma. There is an involution c : L→ L, L = L1 +φ L2, c|Li= ci, i = 1, 2, if and only

if the anti-isomorphism φ is equivariant. In this case, such an involution c is unique.

5.4.5. Proposition. Assume that L = L1 +φ L2, φ : K1 → K2, where K2 splits off as a directsummand, discrL2 = K2 + G2. Fix an involution c : L → L with ci = c|Li

, and assume thatL2 is c2-equivariant D-relative epistable for some D ⊂ L2. Let f1 : L1 → L1 be a c1-equivariantautomorphism inducing (f1)∗ : discrL1 → discrL1 such that (f1)∗(K1) = K1. Then:

(1) f1 can be extended to a c-equivariant automorphism f ∈ Aut(L,D).(2) For any given automorphism φG2 : G2 → G2, the automorphism f in (1) can be chosen so

that its restriction, f2 : L2 → L2 induces φG2 on G2.(3) Given another equivariant anti-isomorphism φ′ : K1 → K2, with the induced from c1 and

c2 involution c′ on L′ = L1 +φ′ L2, there exists an extension of f1 to an isomorphismf : L→ L′, which commutes with c and c′, whose restriction, f2 : L2 → L2 induces on G2

automorphism φG2 , and for which f(D) = D.

Proof. In the parts (1) and (2), we consider an automorphism (f2)∗ on discrL2 which is φ (f1)∗ φ−1 on K2 and φG2 on G2. Equivariant D-relative epistability of L2 implies that (f2)∗can be lifted to an automorphism f2 : L2 → L2 such that f2(D) = D. The anti-isomorphism φ isequivariant, so, by Lemma 5.4.4 the required f exists.

The part (3) is similar, except that we need to start with an involution φ′ (f1)∗ φ−1 onK2.

5.5. Realizability of T-pairs by geometric involutions. In this subsection we prove exis-tence of a geometric involution c ∈ C(L,∆, h) on a (∆, h)-decorated K3-lattice L whose pair ofeigenlattices (T+(c), T−(c)) is isomorphic to a given T-pair (T1, T2). Throughout the subsectionthis pair is supposed to be ascending, although the case of descending T-pairs is analogous. First,we show that T1 and T2 after an appropriate gluing give a lattice isomorphic to T ⊂ L.

5.5.1. Proposition. For any ascending T-pair (T1, T2), there exists a subgroup K2 ⊂ discr2 T2anti-isomorphic to discr2 T1. Any such subgroup is the orthogonal complement K2 = v⊥ of somev ∈ discr2(T2), such that qT2(v) = − 1

2 .

5.5.2. Lemma. Assume that (T1, T2) is an ascending T-pair such that δ2(T1) = 0, and v ∈discr2 T2 is the characteristic element. Then qT2(v) = − 1

2 .


Proof. Lemma 4.9.5 gives Br2(T1) +Br2(T2) = −1, and by additivity Br2(T2) = Br〈v〉+Br(v⊥),where 〈v〉 ⊂ discr2 T2 is spanned by v, while v⊥ is its orthogonal complement. Since the quadraticforms in discr2 T1 and v

⊥ are even, their Brown invariants are divisible by 4, and thus Br〈v〉 = −1mod 4. This implies that qT2(v) = − 1

2 .

5.5.3. Lemma. Assume that (G, q) is an elementary enhanced 2-group, which is not isomorphicto n〈12 〉 for n 6 3, and has δ2(G) = 1 (i.e., q is odd). Then there exists an element v ∈ G, such

that q(v) = − 12 . If in addition G 6= 〈− 1

2 〉, then such v can be chosen non-characteristic.

Proof. Existence of v such that q(v) = − 12 follows from decomposability G = p〈12 〉+ q〈− 1

2 〉 (seeTheorem 3.6.3) and isomorphism 4〈12 〉 ∼= 4〈− 1

2 〉. If p + q > 1, then a generator of a summand

〈− 12 〉 in such a decomposition is non-characteristic.

Proof of Proposition 5.5.1. By Lemma 4.9.6, it is sufficient to prove existence of v ∈ discr2(T2)satisfying two conditions: first, qT2(v) = − 1

2 , and second, v is characteristic if δ2(T1) = 0, andnon-characteristic otherwise. Lemma 5.5.2 proves it in the case δ2(T1) = 0. If δ2(T1) = 1, thenwe can use Lemma 5.5.3 after showing that discr2 T2 cannot be isomorphic to n〈12 〉, 1 6 n 6 3,

or to 〈− 12 〉, if δ2(T1) = 1.

Indeed, if we suppose that discr2 T2 = n〈12 〉, then we have r2(T2) = Br2(T2) = n mod 8,and thus, r2(T1) = n − 1 and due to Lemma 4.9.5, Br2(T1) = −n − 1 mod 8. On the otherhand, if δ2(T1) = 1, then discr2 T1 is isomorphic to p〈12 〉 + q〈− 1

2 〉, p, q > 0, and p + q = n − 1,Br2(T1) = p − q = −n − 1 mod 8, which is impossible for 1 6 n 6 3. If we suppose thatdiscr2 T2 = 〈− 1

2 〉, then we have r2(T1) = r2(T2) − 1 = 0, which contradicts to the assumptionthat δ2(T1) = 1.

5.5.4. Proposition. Any ascending T-pair (T1, T2) is isomorphic to the pair of eigenlattices(T+(cT ), T−(cT )) of some involution cT in the lattice T .

Proof. By Proposition 5.5.1 an anti-isomorphism K1 = discr2 T1φ−→ K2 ⊂ discr2 T2 does exist,

which gives due to Proposition 3.11.1, an involution, c, in T1 +φ T2 with the eigenlattices T1,T2. So, it is left to verify that T1 +φ T2 is isomorphic to T = U + U(3) + 2A2 + A1 ⊂ L (seeLemma 4.6.1). This follows from Nikulin’s stability criterion 3.12.1 applied to T1 +φ T2 andT , because both lattices have the same inertia indices and isomorphic discriminants. For thelatter, note that discr3(T1 +φ T2) = discr3 T1 + discr3 T2 = 〈23 〉 + 3〈− 2

3 〉 = discrT , and that

discr2(T1 +φ T2) = 〈− 12 〉 = discr2 T .

5.5.5. Proposition. Any involution cT : T → T can be extended to a geometric involutionc ∈ C(L,∆, h).

Proof. First, let us extend a given involution cT : T → T to the lattice T ′ = T +φh〈2〉 glued

along the anti-isomorphism φh between discr2 T = 〈− 12 〉 and 〈12 〉 = discr〈2〉, so that the generator

h ∈ 〈2〉 is sent to −h (see Lemma 5.4.4). Next, we extend the involution cT ′ : T ′ → T ′ that weobtain to L as follows.

Let φ : discrS0 → discrT ′ be the anti-isomorphism defined by the sublattices S0, T ′ ⊂ L,L = S0 +φ T

′. Consider the pull-back cdiscrS : discrS0 → discrS0 of the involution induced bycT ′ in discrT ′ via φ. According to Corollary 5.3.3, involution cdiscrS can be lifted to a involution

cS ∈ Aut(S0,∆). Then, cT ′ and cS are compatible and thus yield an involution, c, in L (seeLemma 5.4.4) which is geometric.

Finally, we obtain the following theorem as a direct corollary of Propositions 5.5.4 and 5.5.5.

5.5.6. Theorem. Any ascending T-pair (T1, T2) is geometric.

Remark. As was mentioned in the beginning of previous Subsection, the case of descending T-pairs is analogous, so such pairs are also geometric. ⊡


6. Deformation classes via periods

Through all this section we fix a K3-lattice L and its conical (∆,h)-decoration. Note that anyother conical (∆, h)-decoration can be identified with the fixed decoration via an isomorphism,which exists due to Proposition 4.6.4.

6.1. The complex period map. Recall that, according to Proposition 4.5.2, the resolution

decoration of the K3-surface Y associated with a Zariski sextic is conical. Turning this propertyinto a definition we extend it to any six-cuspidal K3-surface Z which can be equipped with

a lattice isomorphism φ : H2(Z) → L such that the exceptional divisors of the six cusps aremapped to ∆ and the preimage of h belongs to the closure of the cone generated by ampledivisors (note that not any six-cuspidal K3-surface has these properties, cf., [Z1] and [D1]).By an (L,∆, h)-premarked K3-surface we mean any six-cuspidal K3-surface Z equipped witha lattice isomorphism having the above properties, the latter is called an (L,∆, h)-premarking.

Thus, for K3-surfaces Y associated with Zariski sextics, an (L,∆, h)-premarking of Y is nothing

but a lattice isomorphism f : H2(Y ) → L identifying the resolution decoration with the referenceconical decoration of L.

Assume that φ : H2(Z) ∼= H2(Z) → L is an (L,∆, h)-premarking of a six-cuspidal K3-surface

Z. Then the holomorphic 2-forms in Z form a line φ(H2,0(Z)) ⊂ LC = L ⊗ C called the period

line of Z. This line is orthogonal to the sublattice S(∆) ⊂ L (that is the primitive closure ofthe sublattice generated by ∆ and h) and thus lies in TC = T ⊗ C. Taking the projectivization

P (TC) ⊂ P (LC) of TC ⊂ LC we obtain a point Ω = Ω(Z) ∈ P (TC) called the period point (or

simply the period) of Z.More generally, one can define an (L,∆, h)-premarking of a holomorphic, or continuous, family

of six-cuspidal K3-surfaces (for example, associated with a holomorphic family of Zariski sextics)as a locally trivial family of premarkings, and given such a family of premarkings one obtains awell defined holomorphic, or continuous, family of period points in P (TC).

According to Hodge-Riemann bilinear relations, the period point Ω belongs to the quadric

Q = w2 = 0 ⊂ P (TC) and, more precisely, to its open subset D = w ∈ Q |ww > 0. Thissubset has two connected components, which are exchanged by the complex conjugation (this

reflects also switching from the given complex structure on Z to the complex-conjugate one andmultiplying the marking by −1). Writing w ∈ Ω ⊂ TC as w = u+ iv, u, v ∈ TR = T ⊗R, we canreformulate the conditions w2 = 0, ww > 0 as u2 = v2 > 0, uv = 0, which implies that the realplane 〈u, v〉 ⊂ TR spanned by u and v is positive definite and bears a natural orientation u ∧ vgiven by u = Rew, v = Imw. Note that the orientation determined similarly by the conjugatecomplex line Ω ⊂ TC is the opposite one.

The orthogonal projection of a positive definite real plane in TR onto another one is non-

degenerate. Thus, to select one of the two connected components of D we fix an orientation ofpositive definite real planes in TR so that the orthogonal projection preserves it. We call it theprescribed orientation and define an (L,∆, h)-marking as an (L,∆, h)-premarking for which the

orientation u ∧ v of φ(H2,0(Z)) defined by the pairs u = Rew, v = Imw for w ∈ φ(H2,0(Z))is the prescribed one. We denote this component by D and call it the period domain. By theperiod mapping we understand the mapping from the set of (L,∆, h)-marked K3-surfaces to Drespecting the above conventions.

By Aut+(L,∆, h) we denote the group of those automorphisms of the triple (L,∆, h) that pre-serve the prescribed orientation (and thus preserveD). The complementary coset Aut−(L,∆, h) =Aut(L,∆, h) r Aut+(L,∆, h) consists of automorphisms exchanging the connected components

of D.Let us call an (L,∆, h)-marked K3-surface Z regular, if there is no v ∈ L r S(∆) such that

v2 = −2 and vh = vΩ(Z) = 0. The following statement is well known and follows, for example,


from [SD].

6.1.1. Proposition. If Z is a regular (L,∆, h)-marked K3-surface, then the linear system de-fined by h induces a degree 2 map Z → P 2 and this map is a double covering of P 2 branchedover a Zariski sextic. Moreover, this correspondence establishes an isomorphism between thespace of regular (L,∆, h)-marked K3-surfaces Z and the space of (L,∆, h)-marked K3-surfaces

Y associated with Zariski sextics.

Remark. By Riemann-Roch inequality, the homology classes h − ∑ e′′i +2e′i3 and h − ∑ e′i+2e′′i

3(cf., 4.4.1) are realized in Z by effective divisors, and the conic through the cusps making theramification sextic to be a Zariski sextic is nothing but the image of each of these divisors. ⊡

Part (1) of the following theorem is a standard consequence of surjectivity of the periodmapping (see [Ku2]) and Proposition 6.1.1; part (2) follows from the strong Torelli theorem (see[BR]) (a systematic study of plane sextics with arbitrary homological conditions on collectionsof simple singularities via such an approach is undertaken in [D2]).

6.1.2. Theorem. Assume that Ω ⊂ LC is a line orthogonal to h and ∆. Then:

(1) A point Ω ∈ D is the period of a surface Y associated with some Zariski sextic A ⊂ P 2

and some (L,∆, h)-marking φ : H2(Y ) → L if and only if there is no v ∈ Lr S(∆) suchthat v2 = −2 and vh = vΩ = 0.

(2) If we are given another Zariski sextic A′ with an (L,∆, h)-marking φ′ : H2(Y ′) → L

having the same period Ω, then A with A′ are projectively equivalent, and in particular,the projections of Y and Y ′ to P 2 are projectively equivalent.

By this theorem, the image of the period mapping is the complement in D of a certain ar-rangement H of hyperplane sections. As we check below, this arrangement splits naturallyin three sub-arrangements corresponding to three different kinds of codimension one degenera-tions: appearance of a node, gluing of two A2-singularities to an A5-singularity, and degener-ation to a double ruled surface. In arithmetical terms, such a splitting of the arrangement His given by distinguishing three sets of (−2)-roots: V n

2 = v ∈ T | v2 = −2, v 6= h mod 2L,V h2 = v ∈ T | v2 = −2, v = h mod 2L, and V g

2 = v ∈ L | v2 = −2, vh = 0, vS0 6= 0, v 6= hmod 2L. We denote by Hv the hyperplane section of D defined by xv = 0 and, finally, de-fine H to be the union of three arrangements, H = Hn ∪ Hh ∪ Hg, where Hn =

⋃v∈V n

2Hv,

Hh =⋃

v∈V h2Hv, and Hg =

⋃v∈V

g2Hv.

Remark. Note that the hyperplane xv = 0 does not intersect D if v ∈ L, v2 = −2, vh = 0, vS0 6= 0,and v = h mod 2L. Indeed, in such a case the orthogonal complement to the plane generatedby v and e ∈ S0 with e2 = −2, ev > 2 has the positive inertia index < 3, while if the orthogonal

complement contains a point ω ∈ D then it contains a positive definite 3-plane generated by h,Reω, Imω. By this reason we do not need to consider the set v ∈ L | v2 = −2, vh = 0, vS0 6=0, v = h mod 2L, and even could replace V g

2 by v ∈ L | v2 = −2, vh = 0, vS0 6= 0. ⊡In this notation, Theorem 6.1.2 can be rephrased as follows: the set of periods of the K3-

surfaces of Zariski sextics is the complement D r H. Furthermore, together with the strongTorelli theorem it implies the following statement, which is also well known.

6.1.3. Theorem. The space D r H is a fine moduli space of regular (L,∆, h)-marked K3-surfaces. Its quotient by Aut+(L,∆, h) is naturally identified with the space of projective classesof Zariski sextics.

Codimension 1 degenerations of regular (L,∆, h)-marked K3-surfaces are represented by non-singular points ofH. Speaking more formally, such a degeneration is represented by a holomorphicdisc, f : D2 → D, intersecting H transversally at a single point, say, f(0) ∈ H. This gives rise


to a holomorphic family Yt, t ∈ D2, of K3-surfaces presented as double planes Yt → P 2 ramifiedalong Zariski sextics At, experiencing a certain degeneration at t = 0.

6.1.4. Proposition. Consider a codimension 1 degeneration of regular (L,∆, h)-marked K3-

surfaces Yt, t ∈ D2, with the periods f(t) ∈ D degenerating to f(0) ∈ Hv, for some (−2)-root v.

Then the type of the degeneration of Yt and of the corresponding Zariski sextics, At, depends onv as follows.

(1) If v ∈ V n2 , then Yt as well as At experiences a nodal degeneration. And any nodal

degeneration of At and Yt can be represented via such a family.

(2) If v ∈ V g2 , then a pair of cusps of At as well as a pair of cusps of Yt, experiences a

degeneration to A5-singularity. And any codimension one degeneration in which at leastone of the 6A2 experiences a degeneration to a deeper singularity can be represented viasuch a family.

(3) If v ∈ V h2 , then At degenerates into a triple conic and Yt degenerates into an elliptic

K3. Moreover, this elliptic K3 is a double of the ruled surface Σ4 = PP 1(OP 1(4)⊕OP 1)branched along a union of the (−4)-section and a 6-cuspidal trigonal curve in Σ4 whereall the six cusps are coplanar, that is belong to a section of Σ4 disjoint from the (−4)-section. And any degeneration of regular (L,∆, h)-marked K3-surfaces to such a doubleΣ4 and of Zariski sextics to a triple conic can be represented via such a family.

Proof. According to Saint-Donat’s results on the projective models of K3-surfaces, see [SD], thosesurfaces that carry a numerically effective divisor of degree 2 generating the linear system withoutfixed components are the double covers of the plane branched in sextics with simple singularities,and those that carry a numerically effective divisor of degree 2 generating the linear system witha fixed component are the double covers of Σ4 branched along a curve with simple singularities,where the curve is linear equivalent to 2c1(Σ4) and splits in a union of the (−4)-section E0

and a trigonal curve 3E0 + 12F disjoint from E0 (here, F stands for the fiber of Σ4; note thatc1(Σ4) = 2E0 + 6F ). The coplanar condition in statement (3) reflects (and is equivalent to) theexistence of a 6-cuspidal divisor sublattice and a master element in its extension; indeed, theproof of the sufficiency of coplanarity is literally the same as the proof of Lemma 4.4.1, whilethe necessity follows then from transitivity of Aut+(L,∆, h) action on the set of V h

2 -hyperplanes.Thus, there remain to examine codimension one degenerations of Zariski sextic to a sextic withsimple singularities. As is well known and, for example, can be easily deduced from [Pho], theonly codimension one degenerations of Zariski sextics to a sextic with simple singularities arenodal degenerations and A5-degenerations of a pair of cusps, 2A2 ⊂ A5.

Arithmetically, each A5-degeneration of a pair of cusps results in a primitive embedding of2A2 into A5. When 2A2 is primitively embedded into A5, then, since |A2| = 3 and |A5| = 6,the orthogonal complement of 2A2 in A5 is 〈−6〉. Thus, if v ∈ V g

2 then a natural (−6)-vectorappears: it is nothing but the orthogonal projection of 3v on T . We denote the set of all these(−6)-vectors by V6.

6.1.5. Proposition. The set V6 coincides with u ∈ T |u2 = −6, uT ⊂ 3Z.Proof. Necessity follows immediately from the definition of V6. So, let us assume that u ∈T, u2 = −6, and uT ⊂ 3Z. Then, [u3 ] is an element of square − 2

3 in the 3-primary discriminant

component discr3(T ). Thus, [u3 ] is glued to an element of square 23 in discr3(S0). By Lemma

5.1.2, there exist 30 such elements [bij ] = [e′i+2e′′i

3 +2e′j+e′′j

3 ], 1 6 i, j 6 6, i 6= j, [bij ] = −[bji ].

Gluing of u with, say, [bji ] produces a (−2)-root e = 13 (u−(e′i+2e′′i +2e′j+e

′′j )) ∈ L that generates

A5 ⊂ L together with e′i, e′′i , e

′j , e

′′j ; namely, e represents the “middle” root of A5, or equivalently,

u = e′i + 2e′′i + 3e+ 2e′j + e′′j . Note, that eS0 6= 0 mod 2Z and therefore e belongs to V g

2 .


To each v ∈ V n2 ∪ V h

2 we associate the (−2)-reflection ρTv : T → T, x 7→ x + 〈x, v〉v, and toeach u ∈ V6 the (−6)-reflection ρTu : T → T, x 7→ x + 1

3 〈x, u〉u. The first one is evidently therestriction to T of the (−2)-reflection ρv : L → L, while the second one is the restriction to T ofthe reflection ρLu : L → L in the 3-dimensional mirror generated by the triple e, e′i + e′′j , e

′j + e′′i

like in the proof of Proposition 6.1.5. Note also that for each u ∈ V6, the hyperplane section Hu

of D defined by xu = 0 coincides with He we used in our definition of Hg.

6.2. The real period map. Let us fix a geometric involution c ∈ C(L,∆, h) (see Section 4.8),extend c to a complex linear involution on L ⊗ C and denote also by c the induced involutions

on TC, P = P (TC), and D. An (L,∆, h)-marking φ of a real K3-surface (Z, conj) is called c-realif c φ = φ conj. If a real K3-surface (Z, conj) can be equipped with a c-real (L,∆, h)-marking,we say that this surface is of homological type c. A real Zariski sextic is said to be of homological

type c, if the K3-surface Y associated to it is equipped with a real structure conj lifted from the

real structure of P 2 and (Y , conj) is equipped with a c-real (L,∆, h)-marking.

The following Lemma shows that this definition of homological type concords with the onegiven in Section 4.8.

6.2.1. Lemma. If (Z, conj) is of homological type c, and c′ represents the same element inC[L,∆, h] as c, then (Z, conj) is of homological type c′ as well.

Proof. If c = fc′f−1 with f ∈ Aut−(L,∆, h), then c = gc′g−1 where g = cf ∈ Aut+(L,∆, h).

The involution c permutes the two components, D and D, of D, and thus c(D) = D, where

c : TC → TC is the composition of c with the complex conjugation in TC. Let DcRand Dc

Rdenote

the fixed point sets of c restricted to D and D. The latter fixed point set consists of the linesgenerated by w = u+ + iu− such that u± ∈ T± ⊗ R, u2+ = u2− > 0, and the orientation u+ ∧ u−is the prescribed one. Since c is geometric, both Dc

Rand its (trivial) double covering Dc

Rare

nonempty.

As it follows from definitions, the period of a c-real (L,∆, h)-marked K3-surface belongs toDc

R= x ∈ D | c(x) = x. Therefore, we call Dc

Rthe real period domain. It splits in a direct

product,

DcR = D(T+)×D(T−),

where D(T+) = D∩P (T+⊗R) and D(T−) = D∩P (T−⊗R) are the real hyperbolic (Lobachesvki)spaces associated with the (hyperbolic) lattices T±. In other words, one can fix a half of the coneu2+ > 0 in T+ ⊗ R and a half of the cone u2− > 0 in T− ⊗ R in a way that the prescribed halvesrespect the prescribed orientation (that is to make the orientation u+ ∧ u− to be the prescribedorientation for any u+, u− from the fixed half-cones) and then D(T±) become the spaces of thevector half-lines in the chosen two half-cones.

6.2.2. Theorem. The periods of c-real regular (L,∆, h)-marked K3-surfaces form in DcRthe

complement of DcR∩ H. The space Dc

Rr H is a fine moduli space of c-real regular (L,∆, h)-

marked K3-surfaces.

Proof. This is a straightforward consequence of Theorems 6.1.2 and 6.1.3.

Let us put V n2 (T±) = V n

2 ∩ T±, V h2 (T±) = V h

2 ∩ T±, V6(T±) = V6 ∩ T± and define Hn(T±) =⋃v∈V n

2 (T±)Hv ∩ D(T±), Hh(T±) =⋃

v∈V h2 (T±)Hv ∩ D(T±), Hg(T±) =

⋃u∈V6(T±)Hu ∩ D(T±).

We denote by Ch± the set of connected components of the complement D(T±) r H± of thehyperplane arrangement H± = Hn(T±)∪Hh(T±)∪Hg(T±). Every of these (infinite in number)components is obviously a convex polyhedron.


6.2.3. Lemma. Every connected component of DcRrH is obtained from a product P+ × P− of

two polyhedra P± ∈ Ch± by removing a codimension two subset. In particular, the inclusion mapidentifies the set of connected components of Dc

RrH with Ch+ × Ch−.

Proof. The difference is due to (−2)- and (−6)-vectors v ∈ V n2 ∪ V h

2 ∪ V6, v /∈ T±, that havenonzero components v± ∈ T± ⊗R and hence Hv ∩Dc

R= v⊥+ × v⊥− (orthogonal complement v⊥± is

in D(T±)) is of codimension 2.

By Aut+(L,∆, h, c) we denote the subgroup of Aut+(L,∆, h) consisting of those elements ofthe latter that commute with c. The important examples of such elements are ρv with v ∈V n2 (Tǫ) ∪ V g

2 (Tǫ) and ρLu with u ∈ V6(Tǫ), ǫ = ±. Note that all these elements act as a reflection

in Tǫ and as the identity in T−ǫ.Each element of Aut+(L,∆, h, c) preserves the eigenspaces T± and hence, group Aut+(L,∆, h, c)

naturally acts on Ch+,Ch−, and Ch+ × Ch−.

6.2.4. Theorem. The set of deformation classes of real Zariski sextics of homological type c isin a natural bijection with the set of orbits of the action of Aut+(L,∆, h, c) on Ch+ × Ch−.

Proof. Given a real Zariski sextic of homological type c, we replace it by the associated K3-

surface Y and equip Y with a real structure conj (lifted from the real structure of P 2) and witha c-real (L,∆, h)-marking. Due to Proposition 6.1.1, the sextic can be reconstructed back, upto projective transformation, as the branch curve of the linear system given by h. Furthermore,if we start from a regular c-real (L,∆, h)-marked K3-surface Z and apply this reconstructionprocedure, the real structure we get on P 2 is the standard one, since all real structures on P 2

are isomorphic to each other. Now, it remains to notice that the c-real (L,∆, h)-markings of Yform an orbit of the action of Aut+(L,∆, h, c) as it follows from Theorem 6.2.2, Lemma 6.2.3,and the strong Torelli theorem.

6.3. Deformation classification via geometric involutions.

6.3.1. Theorem. For any homological type c ∈ C(L,∆, h) there is one and only one deforma-tion class of real Zariski sextics of homological type c. In particular, the deformation classes ofreal Zariski sextics are in one-to-one correspondence with the set of conjugacy classes of ascendinggeometric involutions.

Let us fix a geometric involution c ∈ C(L,∆, h).

6.3.2. Proposition. The action of Aut+(L,∆, h, c) on Ch+ × Ch− is transitive.

Let us analyze how the reflections ρTv for v ∈ V (T±) can be extended from T± to the wholeL. The extension in case of v ∈ V2(T±) is obvious.

6.3.3. Lemma. For any v ∈ V2(T±), the reflection ρv(x) = x − xvv2 v is well-defined on the

whole lattice L, it acts as the identity in S as well as in T∓ (opposite to T±), and ρv ∈Aut+(L,∆, h, c).

The case of v ∈ V6(T±) is a bit more subtle.

6.3.4. Lemma. For any v ∈ V6(T±), the reflection ρTv |T± : T± → T± can be extended to an

automorphism f ∈ Aut+(L,∆, h, c) that acts on T∓ as idT∓ .

Proof. Note that ρTv |T± induces the identity map in discr2 T± and, thus, by Proposition 3.11.1can be extended to the whole T by gluing with the identity automorphism on T∓. Applying

Proposition 3.11.1 once more, we extend the result of gluing to an involution ρT′

v : T ′ → T ′ that

maps h to h. Finally, we extend ρT′

v to f : L → L applying Proposition 5.4.5, where we use the c-equivariant ∆-relative epistability of S0 (see Corollary 5.3.4). To check that f ∈ Aut+(L,∆, h, c)


it is left to notice that ρTv |T± preserves the halves of the cones v ∈ T± | v2 > 0, and thereforef preserves the prescribed orientation of the positive definite real planes in TR.

Proof of Proposition 6.3.2. The reflection group generated by ρTv , v ∈ V n2 (T±)∪V g

2 (T±)∪V6(T±),acts transitively on Ch±. Thus, Lemmas 6.3.3 and 6.3.4 imply Proposition 6.3.2.

Proof of Theorem 6.3.1. The ”only if” part is trivial. The other part follows from Theorem 6.2.4and Proposition 6.3.2.

6.4. From a trigonal curve in Σ4 to Zariski sextics in P 2. With a given geometric ascend-ing involution c ∈ C<(L,∆, h) and an element v ∈ V h

2 (T−) we associate its partner involutioncv that acts in T as a composition of c with reflection against the plane generated by h and v,and in S0 as a composition of c with the involution permuting the elements of ∆ so that thegenerators in each of the six A2 ⊂ S0 are transposed. Then cv ∈ C<(L,∆, h) as well, since bythe definition T+(c

v) = T−(c) ∩ v⊥ and T−(cv) ∩ v⊥ = T+(c).

We can peak a c-real (L,∆, h)-marked K3-surface Z, whose period point is a generic pointof Hv. Then according to Proposition 6.1.4(3), it is a double covering of Σ4 branched alongthe union of the (−4)-section and a 6-cuspidal trigonal curve with coplanar cusps. In additionto the real structure in Z inducing involution c in its lattice, there is another one inducing cv;the two real structures in Z are the two liftings of a real structure in Σ4 and differ by the decktransformation of the double covering Z → Σ4. By real perturbations of (Z, c) and (Z, cv) weobtain regular (L,∆, h)-marked K3-surfaces and the deformation types of the corresponding realZariski sextics do not depend on the choice of perturbations. Hence, we may speak on the welldefined partner Z-deformation classes of real Zariski sextics. Below, we make some explicit choiceof perturbations, which allows us to compare the partner real Zariski sextics obtained in this way.Namely, we prove that such a pair of real Zariski sextics is placed in RP 2 in a trigonal reverseposition (see Section 2.6 for definitions of reverse position, trigonality, and reversion partners).

6.4.1. Proposition. Real Zariski curves in a pair of partner Z-deformation classes are rever-sion partners.

Proof. Consider the weighted projective space P (2, 1, 1, 1) with coordinates q, x, y, z of weights2, 1, 1, 1 and the conic B defined by equations q = 0, Q = 0 where Q denotes the polynomialxy − z2. Note that the cone over B with vertex at (1, 0, 0, 0) is the ruled surface Σ′

4 (the (−4)section of Σ4 is contracted here to the vertex of the cone). Note also that due to the isomorphismP 1 → B given by x = u2, y = v2, z = uv any homogeneous degree 2n polynomial g2n(u, v) invariables u, v can be rewritten as a homogeneous degree n polynomial fn(x, y, z) in x, y, z (sucha presentation is unique modulo the ideal generated by Q).

Now, pick a 6-cuspidal trigonal curve C in Σ′4 so that all the six cusps belong to the plane

q = 0 and write a defining polynomial of C as q3+g4(u, v)q2+g8(u, v)q+g12(u, v). Then, express

the polynomials g4, g8, g12 in variables u, v as polynomials f2, f4, f6 in variables x, y, z. Thisallows us to define C ⊂ Σ′

4 by equations

q3 + f2(x, y, z)q2 + f4(x, y, z)q + f6(x, y, z) = 0, Q = 0,

and then include it in a family

q3 + f2(x, y, z)q2 + f4(x, y, z)q + f6(x, y, z) = 0, Q = tq.

Due to the special position of the cusps q3 + g4(u, v)q2 + g8(u, v)q+ g12(u, v) = q3 + (g2(u, v)q+

g6(u, v))2, so that q3 + f2(x, y, z)q

2 + f4(x, y, z)q + f6(x, y, z) = q3 + (f1(x, y, z)q + f3(x, y, z))2.

As a straightforward calculation shows, the associated family of plane curves defined by

Q3 + t(f1(x, y, z)Q+ tf3(x, y, z))2 = 0


is a family of Zariski sextics (with 6 cusps on B) degenerating to Q3 = 0. The double coveringsof P 2 branched along these sextics are naturally embedded into the weighted projective spaceP (3, 2, 1, 1, 1) with coordinates r, q, x, y, z of weights 3, 2, 1, 1, 1 where they become defined byequations

r2 + (q3 + (f1(x, y, z)q + f3(x, y, z))2) = 0, Q = tq,

and thus form a family converging to the double covering of Σ4 branched along C:

r2 + (q3 + (f1(x, y, z)q + f3(x, y, z))2) = 0, Q = 0.

To get a pair of real Zariski curves in a pair of Z-deformation classes we need to select a pairof Mobius real structures converging to (Z, c) and (Z, cv). In other words to select appropriatereal forms of the double coverings, namely, to choose an appropriate sign of t and an appropriatesign of r2. Such a pair of real forms is given, with respect to the standard real structure onP (3, 2, 1, 1, 1), by

r2 + (q3 + (f1(x, y, z)q + f3(x, y, z))2) = 0, Q = tq,

for t > 0 and by

r2 − (q3 + (f1(x, y, z)q + f3(x, y, z))2) = 0, Q = tq,

for t < 0. The real Zariski curves C′t, t > 0, of the first family converge to C, while the real

Zariski curves C′′t , t < 0, of the second one converge to the image C′ of C under the reflection

q 7→ −q in the cylindrical part of Σ′4, and thus the trigonality and reversion position property

from the definition of reversion partners follows from the possibility to trivialize the real part ofthe surface family over the real locus of the cylindrical part of Σ′

4. The signs o(C) and o(C′) areopposite due to Corollary 4.6.3, while Proposition 4.3.5 implies that C and C′ are both of thesame type.

7. Arithmetics of the T-pairs

7.1. Geography of the ascending T-pairs. We start analysis of the T-halves, which wedenote here by M , with reviewing their possible numerical invariants. On the first step, weignore its 3-primary component and look only at the rank r(M), the discriminant 2-rank r2(M),and the discriminant parity δ2(M). Namely, we prove the following

7.1.1. Lemma. Assume that (M,M ′) is an ascending T-pair. Then the combination of theinvariants r and r2 of M is one of the fifteen ones listed in the Table 3A, and the combinationof the invariants r′ = 9 − r, r′2 = r2 + 1 of M ′ appears at the corresponding position in Table3B. The value of δ2 for M indicated in each row of Table 3A is the only possible value for allthe pairs (r, r2) in this row (for the last row, the both values, 0 and 1, of δ2 are allowable). Thevalue δ′2 for M ′ is 1 for each pair (r′, r′2).

Table 3A. T-halves M

(r, r2) δ2————————— —

(2, 0), (4, 0), (6, 0), (8, 0) 0(1, 1), (3, 1), (5, 1), (7, 1) 1

(3, 3), (5, 3) 1(2, 2), (4, 2), (6, 2), (4, 4) 0, 1

Table 3B. T-halves M ′

(r′, r′2) δ′2————————— —

(7, 1), (5, 1), (3, 1), (1, 1) 1(8, 2), (6, 2), (4, 2), (2, 2) 1

(6, 4), (4, 4) 1(7, 3), (5, 3), (3, 3), (5, 5) 1


Proof. For T-pairs the relations 1 6 r, r′ 6 8, r + r′ = 9 hold by definition, the ascendingcondition means r′2 = r2 + 1, and Lemma 4.9.1 yields r2 6 min(r, r′ − 1). Together with thecongruence r2 = r mod 2 (see Lemma 3.4.2), these restrictions forbid all the cases except thefourteen ones listed in the Tables 3A-B.

The value δ′2 = 1 follows from the definition of ascending T-pairs in 4.9 (property (4) in Lemma4.8.1). For the values of δ2 in Table 3A note that for r2 = 0 we have discr2M = 0, and thusδ2 = 0. On the other hand, for odd values of r2, we have δ2 = 1, see Theorem 3.6.3(1).

Now, for each combination of values r, r2, δ2 from Tables 3A-B we will finalize the enumerationof the IDs by giving possible values of (p, q) describing the discriminant 3-component, discr3M =p〈23 〉+ q〈− 2

3 〉. Namely, we prove the following.

7.1.2. Proposition. Assume that (M,M ′) is an ascending T -pair. Then the combination ofthe invariants r, r2, δ2, p, q for M and the corresponding invariants r′ = 9 − r, r′2 = r2 + 1,p′ = 1− p, q′ = 3− q for M ′ (recall that δ′2 = 1) is contained in one or the rows of Table 4 (thematching values of (p, q) and (p′, q′) appear in the corresponding positions of that row).

Table 4.

δ2 (r, r2) (p, q) (r′, r′2) (p′, q′)

—– —– ————— —— —————0 (2, 0) (0, 0), (1, 1) (7, 1) (1, 3), (0, 2)0 (4, 0) (0, 1), (1, 2) (5, 1) (1, 2), (0, 1)0 (6, 0) (0, 2), (1, 3) (3, 1) (1, 1), (0, 0)0 (8, 0) (0, 3) (1, 1) (1, 0)1 (1, 1) (0, 0), (1, 0) (8, 2) (1, 3), (0, 3)

1 (3, 1) (0, 0), (0, 1), (1, 1), (1, 2) (6, 2) (1, 3), (1, 2), (0, 2), (0, 1)1 (5, 1) (0, 1), (0, 2), (1, 2), (1, 3) (4, 2) (1, 2), (1, 1), (0, 1), (0, 0)1 (7, 1) (0, 2), (0, 3), (1, 3) (2, 2) (1, 1), (1, 0), (0, 0)

0, 1 (2, 2) (0, 0), (1, 1) (7, 3) (1, 3), (0, 2)1 (2, 2) (0, 1), (1, 0) (7, 3) (1, 2), (0, 3)0 (4, 2) (0, 3), (1, 0) (5, 3) (1, 0), (0, 3)

0, 1 (4, 2) (0, 1), (1, 2) (5, 3) (1, 2), (0, 1)1 (4, 2) (0, 0), (0, 2), (1, 1), (1, 3) (5, 3) (1, 3), (1, 1), (0, 2), (0, 0)0 (6, 2) (1, 1) (3, 3) (0, 2)

0, 1 (6, 2) (0, 2), (1, 3) (3, 3) (1, 1), (0, 0)1 (6, 2) (0, 1), (0, 3), (1, 2) (3, 3) (1, 2), (1, 0), (0, 1)1 (3, 3) (0, 0), (0, 1), (0, 2), (1, 0), (1, 1), (1, 2) (6, 4) (1, 3), (1, 2), (1, 1), (0, 3), (0, 2), (0, 1)1 (5, 3) any p 6 1, q 6 3 (4, 4) p′ = 1− p, q′ = 3− q

0, 1 (4, 4) (0, 3), (1, 0) (5, 5) (1, 0), (0, 3)1 (4, 4) (0, 0), (0, 1), (0, 2), (1, 1), (1, 2), (1, 3) (5, 5) (1, 3), (1, 2), (1, 1), (0, 2), (0, 1), (0, 0)

The proof is based on the following relations.

7.1.3. Proposition. Assume that lattice L is even, the discriminant p-ranks rp vanish for allp 6= 2, 3, and the primary components discrp(L) for p = 2 and p = 3 are elementary. Let σ ∈ Z

denote the signature of L, δ2 ∈ 0, 1 the parity of discr2 L, and Br3 the Brown invariant ofdiscr3 L. Then

(1) r2 = 0 implies that Br3 = σ mod 8;(2) r2 = 1 implies that |Br3 − σ| = 1 mod 8;(3) r2 = 2 together with |Br3 − σ| = 4 mod 8 imply that δ2 = 0;(4) δ2 = 0 implies that r2 is even and Br3 = σ mod 4.(5) r2 = r together with δ2 = 0 imply that Br3 = −σ mod 8;(6) r3 = r implies that Br3 = 2σ mod 8.


Proof. The Brown invariant Br2 of discr2 L can be expressed as the modulo 8 residue of σ−Br3by the van der Blij theorem (see Corollary 3.7.4). Since Br2 = 0 for r2 = 0, this implies (1). Ifr2 = 1, then Br2 = ±1 (see Example 3.6.1), which implies (2). The case of r2 = 2 with δ2 = 0 isconsidered in the Example 3.6.2, it yields (3). Theorem 3.6.3(1) yields (4).

If r2 = r, then L is divisible by 2 (see Proposition 3.4.1), moreover the condition δ2 = 0 impliesthat division by 2 yields an even lattice L′ = L(12 ). Moreover, parts (2) and (3) of Proposition3.4.1 imply that discrL′ is anti-isomorphic to discr3 L, and thus, Br(L′) = −Br3(L). Applyingthe van der Blij theorem to L′, we obtain Br(L′) = σ mod 8, and thus Br3(L) = −σ, andapplying it to L, we get Br(L) = Br2(L) + Br3(L) = σ mod 8, and thus, Br2(L) = 2σ mod 8,which yields (5).

If r3 = r, then L is divisible by 3 and L′ = L(13 ) is an even lattice whose discriminantdiscrL′ = discr2 L

′ (again, due to Proposition 3.4.1) is anti-isomorphic to discr2 L. Like in theprevious case, applying van der Blij’s theorem to L′ and L we obtain Br(L′) = −Br2(L) = σmod 8, and Br(L) = Br2(L) + Br3(L) = σ mod 8, which yields (6).

7.1.4. Corollary. Assume that M is an T -half of rank r, 2-rank r2, and discr3M = p〈23 〉 +q〈− 2

3 〉. Then

(1) if r2 = 0, then q − p = r2 − 1 mod 4;

(2) if r2 = 1, then q − p = r±12 − 1 mod 4;

(3) if r2 = 2 and q − p = r2 + 1 mod 4, then δ2 = 0;

(4) if δ2 = 0, then r = r2 = 0 mod 2 and q − p = r2 − 1 mod 2;

(5) if r2 = r and δ2 = 0, then p− q = r2 − 1 mod 4.

(6) if r3 = r, then q − p = r − 2 mod 4;(7) r > r3 = p+ q.

Proof. (1)–(6) follows from Proposition 7.1.3, since for T -halves Br3 = 2(p − q) mod 8 andσ = 2− r. (7) follows from Proposition 3.4.1.

Proof of Proposition 7.1.2. It follows from the definition of T-halvesM (see Lemma 4.8.1(5)) andclassification of the elementary 3-group (see Lemma 3.7.2) that discr3M = p〈23 〉+ q〈− 2

3 〉, where0 6 p 6 1 and 0 6 q 6 3. For each combination of (r, r2) and δ2 from Table 3A, we includedin Table 4 those pairs (p, q) which are not forbidden by one of the restrictions of Corollary 7.1.4applied to M and M’.

Remark. In a few cases Corollary 7.1.4 does not forbid a combination of δ2, r, r2, p, and q, butforbids the complementary combination (i.e., for the other T-half of an T-pair). For instance,the case (r, r2) = (7, 1), (p, q) = (1, 0) is not forbidden, and is in fact realizable for the lattice〈2〉 + E6. However, the complementary lattice would have r′ = 2 and (p′, q′) = (0, 3), which isforbidden, since r′ < r′3 = 3. Thus, the combination (r, r2) = (7, 1), (p, q) = (1, 0) is excludedfrom the list. ⊡

7.2. The list of T-halves. Below we provide an example of a T-half for every combination ofthe numerical invariants from Table 4 (verification is straightforward, cf., [N1], Theorem 1.10.1).

7.2.1. Proposition. For every combination of invariants r, r2, δ2, p, q, or r′, r′2, δ

′2 = 1, p′,

q′ listed in the Table 4, there exists a T-half with such invariants. Namely, such a lattice can befound in Tables 5A-J.

Tables 5A-J contain only T-halves. Sign “-” signifies that a lattice with given invariants isforbidden by Corollary 7.1.4. Sign “*” stands if a lattice actually exists, but it is not a T-half,


because the complementary lattice is forbidden.

Table 5A. r2 = 0 and p = 0

(r, r2) q = 0 q = 1 q = 2 q = 3(2, 0) U − − −(4, 0) − U+ A2 − −(6, 0) − − U+ 2A2 −(8, 0) − − − U+ 3A2

Table 5B. r2 = 0 and p = 1

q = 0 q = 1 q = 2 q = 3− U(3) − −− − U(3) + A2 −− − − U(3) + 2A2

U+ E6 − − −

Table 5C. r2 = 1 and p=0

(r, r2) q = 0 q = 1 q = 2 q = 3(1, 1) 〈2〉 − − −(3, 1) U+ A1 〈2〉 + A2 − −(5, 1) − U+ A2 + A1 〈2〉 + 2A2 −(7, 1) − − U+ 2A2 + A1 〈2〉 + 3A2

Table 5D. r2 = 1 and p=1

q = 0 q = 1 q = 2 q = 3〈6〉 − − −− 〈6〉 + A2 U(3) + 〈−6〉 −− − 〈6〉 + 2A2 U(3) + A2 + 〈−6〉∗ − − 〈6〉 + 3A2

Table 5E. r2 = 2 and p=0

δ2 (r, r2) q = 0 q = 1 q = 2 q = 30 (2, 2) U(2) − − −1 (2, 2) 〈2〉 + A1 〈2〉 + 〈−6〉 − −0 (4, 2) − U(2) + A2 − U(3) + A2(2)1 (4, 2) U+ 2A1 U+ A1 + 〈−6〉 U+ 2〈−6〉 −0 (6, 2) ∗ − U(2) + 2A2 −1 (6, 2) − U+ A2 + 2A1 U+ A2 + 〈−6〉 + A1 U+ A2 + 2〈−6〉1 (8, 2) − − ∗ 〈2〉 + 3A2 + A1

Table 5F. r2 = 2 and p=1

δ2 (r, r2) q = 0 q = 1 q = 2 q = 30 (2, 2) − U(6) − −1 (2, 2) 〈6〉+ A1 〈6〉+ 〈−6〉 − −0 (4, 2) U+ A2(2) − U(6) + A2 −1 (4, 2) − U(3) + 2A1 U(3) + A1 + 〈−6〉 U(3) + 2〈−6〉

0 (6, 2) − U(3) + D4 − U(6) + 2A2

1 (6, 2) ∗ − U(3) + A2 + 2A1 U(3) + A2 + A1 + 〈−6〉1 (8, 2) ∗ ∗ − U(3) + 2A2 + 2A1

Table 5G. r2 = 3, or r2 = 5 and p=0

(r, r2) q = 0 q = 1 q = 2 q = 3(3, 3) 〈2〉+ 2A1 〈2〉 + A1 + 〈−6〉〈2〉 + 2〈−6〉 −(5, 3) U+ 3A1 U+ 2A1 + 〈−6〉 U+ A1 + 2〈−6〉 U+ 3〈−6〉(5, 5) U(2) + 3A1 U(2) + 2A1 + 〈−6〉 U(2) + A1 + 2〈−6〉 U(2) + 3〈−6〉(7, 3) ∗ ∗ U+ A2 + 〈−6〉+ 2A1 U+ A2 + 2〈−6〉 + A1

Table 5H. r2 = 3, or r2 = 5 and p=1

(r, r2) q = 0 q = 1 q = 2 q = 3(3, 3) 〈6〉 + 2A1 〈6〉 + A1 + 〈−6〉〈6〉+ 2〈−6〉 −(5, 3) 〈6〉 + D4 U(3) + 3A1 U(3) + 2A1 + 〈−6〉 U(3) + A1 + 2〈−6〉(5, 5) 〈6〉 + 4A1 U(6) + 3A1 U(6) + 2A1 + 〈−6〉 U(6) + A1 + 2〈−6〉(7, 3) ∗ ∗ U(3) + A2 + 3A1 U(3) + A2 + 〈−6〉+ 2A1

Table 5I. r2 = 4 and p=0

δ2 (r, r2) q = 0 q = 1 q = 2 q = 30 (4, 4) − − − U(6) + A2(2)1 (4, 4) 〈2〉+ 3A1 〈2〉 + 2A1 + 〈−6〉〈2〉 + A1 + 2〈−6〉〈2〉 + 3〈−6〉1 (6, 4) ∗ U+ 〈−6〉+ 3A1 U+ 2〈−6〉 + A1 U+ 3〈−6〉 + A1


Table 5J. r2 = 4 and p=1

δ2 (r, r2) q = 0 q = 1 q = 2 q = 30 (4, 4) U(2) + A2(2) − − −1 (4, 4) 〈6〉 + 3A1 〈6〉 + 〈−6〉 + 2A1 〈6〉 + 2〈−6〉 + A1 〈6〉 + 3〈−6〉1 (6, 4) ∗ U(3) + 4A1 U(3) + 〈−6〉+ 3A1 U(3) + 2〈−6〉 + 2A1

Remark. Note that there is some ambiguity in the direct sum presentations of the lattices in theabove tables, due to the following isomorphisms: 〈6〉 + A2 = U(3) + A1, 〈2〉 + A2 = U + 〈−6〉,〈6〉+ A2(2) = 〈2〉+ 2〈−6〉, 〈2〉+ A2(2) = 〈6〉+ 2A1.

Besides, for any lattice L with an odd discr2 L (for instance, L = A1, or L = 〈−6〉 ) we haveU(2)+L = 〈2〉+A1 +L, and U(6)+L = 〈6〉+ 〈−6〉+L. ⊡

7.3. Stability of the T-halves. Our aim in the next four subsections is to prove stability ofthe T-halves, and thus, to show that their list in Tables 5A-J is complete.

It is easy to check which of these lattices do not satisfy Nikulin’s stability criterion (see 3.12.1)using Table 4. We see from it that Nikulin’s condition for r2, namely, r > r2, or r = r2 > 2, failsonly for the T-halves listed in Table 6A below. Table 6B contains the remaining cases, in whichNikulin’s condition r3 6 r − 2 fails.

7.3.1. Lemma. For any T-pair (T1, T2), Nikulin’s stability criterion 3.12.1 is satisfied for Ti,i ∈ 1, 2, unless the combination of the invariants r(Ti), r2(Ti), δ2(Ti), p(Ti) and q(Ti) is amongthe ones listed in Tables 6A-B.

Table 6A. r2 = r 6 2

12345678

δ2 (r, r2) (p, q) Ti1 (1, 1) (0, 0) 〈2〉1 (1, 1) (1, 0) 〈6〉0 (2, 2) (0, 0) U(2)0 (2, 2) (1, 1) U(6)1 (2, 2) (0, 0) 〈2〉+ A1

1 (2, 2) (0, 1) 〈2〉+ 〈−6〉1 (2, 2) (1, 0) 〈6〉+ A1

1 (2, 2) (1, 1) 〈6〉+ 〈−6〉

7.3.2. Corollary. For any T-pair (T1, T2) either T1 or T2 satisfies Nikulin’s stability condition,and so is stable and epistable.

Proof. Is can be easily seen from Table 4, that there is no pairs (T1, T2) for which both T1 andT2 are represented in the above list of exceptions.

Our next aim is to verify stability of the lattices in Tables 6A-B. The Miranda-Morrisoncriterion (see Proposition 3.12.2) gives the following statement.

7.3.3. Proposition. Assume that M is a T-half of rank r > 2 and that one of the discriminantp-ranks rp, p = 2, 3 of M is less than r. Then M is stable and epistable.

The following fact is well known.

7.3.4. Lemma. The only unimodular hyperbolic lattices of rank 6 8 are U and 〈1〉 + n〈−1〉,n 6 7.


Table 6B. r3 > r − 1

123456789101112131415161718

δ2 (r, r2) (p, q) Ti0 (2, 0) (1, 1) U(3)0 (4, 0) (1, 2) U(3) + A2

1 (3, 1) (1, 1) U(3) + A1

1 (3, 1) (1, 2) U(3) + 〈−6〉1 (5, 1) (1, 3) U(3) + A2 + 〈−6〉0 (4, 2) (0, 3) U(3) + A2(2)0 (4, 2) (1, 2) U(6) + A2

1 (4, 2) (1, 2) U(3) + A1 + 〈−6〉1 (4, 2) (1, 3) U(3) + 2〈−6〉1 (3, 3) (0, 2) 〈2〉+ 2〈−6〉1 (3, 3) (1, 1) 〈6〉+ A1 + 〈−6〉1 (3, 3) (1, 2) 〈6〉+ 2〈−6〉1 (5, 3) (1, 3) U(3) + A1 + 2〈−6〉0 (4, 4) (0, 3) U(6) + A2(2)1 (4, 4) (0, 3) 〈2〉+ 3〈−6〉1 (4, 4) (1, 2) 〈6〉+ A1 + 2〈−6〉1 (4, 4) (1, 3) 〈6〉+ 3〈−6〉1 (5, 5) (1, 3) U(6) + A1 + 2〈−6〉

r3 = rr3 = r − 1r3 = r − 1r3 = r

r3 = r − 1r3 = r − 1r3 = r − 1r3 = r − 1r3 = r

r3 = r − 1r3 = r − 1r3 = r

r3 = r − 1r3 = r − 1r3 = r − 1r3 = r − 1r3 = r

r3 = r − 1

r2 = 0r2 = 0r2 < rr2 < rr2 < rr2 < rr2 < rr2 < rr2 < rr2 = rr2 = rr2 = rr2 < rr2 = rr2 = rr2 = rr2 = rr2 = r

7.3.5. Lemma. Assume that M is a T-half. Let r be its rank and rp, p = 2, 3 its discriminantp-ranks. Then:

(1) If r2 = 0 and r3 = r, then M = U(3).(2) If r2 = r and r3 = 0, then either δ2 = 0 and M = U(2), or δ2 = 1 and M = 〈2〉+ n〈−2〉,

n 6 7.(3) If r2 = r3 = r, then either δ2 = 0 and M = U(6), or δ2 = 1 and M = 〈6〉+n〈−6〉, n 6 7.

Proof. It is enough to apply Proposition 4.9.2 and Lemma 7.3.4 to M(13 ), M(12 ), and M(16 ) inthe cases (1), (2), and (3), respectively (the estimate n 6 7 is due to Proposition 4.9.2(1)).

7.3.6. Lemma. Under the assumption of Lemma 7.3.5, if r2 = r = 2 and r3 = 1, then L iseither 〈2〉+ 〈−6〉, or 〈6〉+ 〈−2〉.

Proof. The lattice L′ = L(12 ) has discriminant order | discrL′| = 3. Fixing any basis of L, we

can present it by a matrix,

(a bb c

), ac − b2 = −3, (the sign of the discriminant is due to that

L is indefinite). The gaussian theory of normal forms says that in a suitable basis this matrix

has 0 6 b 6√3, i.e., b = 0, or b = 1. If b = 1, then ac = b2 − 3 = −2, and the only solutions

(up to reordering the basis) are

(±1 11 ∓2

)and it is easy to check that the both matrices are

diagonalizable. Diagonalization yields

(±3 00 ∓1

), that is 〈1〉+ 〈−3〉, or 〈3〉+ 〈−1〉.

7.3.7. Corollary. The lattices of Table 6A and the lattices of Table 6B having r3 = r, namely,the ones in rows 1,4,9,12, and 17, are all stable.

Proof. Lattices of Table 6A satisfy either assumptions (2)–(3) of Lemma 7.3.5, or the ones ofLemma 7.3.6. For the indicated lattices of Table 6B we apply Lemma 7.3.5.

7.3.8. Theorem. All the lattices in Tables 5A-J are stable.


Proof. The cases of lattices with r 6 2 for which Nikulin’s and Miranda-Morrisson’s criteria donot guarantee stability are covered by Lemmas 7.3.5 and 7.3.6. The case (3) of Lemma 7.3.5covers also r2 = r3 = r > 3. In the remaining cases Proposition 7.3.3 can be applied.

7.3.9. Corollary. Tables 5A-J give a complete list of T-halves.

Proof. Proposition 7.1.2 gives a list of the invariants (r, r2, δ2, p, q) which can be potentiallyrealized for T-halves. Proposition 7.2.1 shows that this list is actually realized by lattices inTables 5A-J. Theorem 7.3.8 shows that there is no other T-halves with the same invariants, andso, Tables 5A-J give a complete list.

7.3.10. Corollary. A T-half M is determined by its rank r, 2-rank r2, parity δ2 of discr2M ,and the characteristics 0 6 p 6 1, 0 6 q 6 3 of discr3M = p〈23 〉 + q〈− 2

3 〉. In particular, M isdetermined by r and discrM .

If we combine T-halves in Tables 5A-J into ascending T-pairs in accordance with the Table 4,then we obtain the following result.

1234567891011121314151617181920212223242526272829303132

Table 7B. Ascending T-pairs T± in the case of p = 1

δ2 (r+, r2) q T+ T−0 (2, 0) 1 U(3) U+ 3A2

0 (4, 0) 2 U(3) + A2 U+ A2 + A1

0 (6, 0) 3 U(3) + 2A2 U+ A1

1 (1, 1) 0 〈6〉 U+ 2A2 + A1 + 〈−6〉1 (3, 1) 1 〈6〉+ A2 〈2〉+ 2A2 + A1

1 (3, 1) 2 U(3) + 〈−6〉 U+ A2 + 2A1

1 (5, 1) 2 U(3) + A2 + A1 U+ 〈−6〉+ A1

1 (5, 1) 3 U(3) + A2 + 〈−6〉 U+ 2A1

1 (7, 1) 3 〈6〉+ 3A2 〈2〉+ A1

0 (2, 2) 1 U(6) U(2) + 2A2 + A1

1 (2, 2) 1 〈6〉+ 〈−6〉 U+ A2 + 2A1 + 〈−6〉1 (2, 2) 0 〈6〉+ A1 〈2〉+ 2A2 + A1 + 〈−6〉0 (4, 2) 0 U+ A2(2) U(3) + A2(2) + A1

0 (4, 2) 2 U(6) + A2 U(2) + A2 + A1

1 (4, 2) 2 〈6〉+ A2 + 〈−6〉〈2〉+ A2 + 2A1

1 (4, 2) 1 〈6〉+ A2 + A1 〈2〉+ 〈−6〉+ A2 + A1

1 (4, 4) 3 〈6〉+ 3〈−6〉〈2〉+ 4A1

1 (4, 2) 3 U(3) + 2〈−6〉 U+ 3A1

0 (6, 2) 3 U(6) + 2A2 U(2) + A1

0 (6, 2) 1 U(3) + D4 〈6〉+ A2(2)1 (6, 2) 2 〈6〉+ 2A2 + A1 〈2〉+ A1 + 〈−6〉1 (6, 2) 3 U(3) + A2 + A1 + 〈−6〉〈2〉+ 2A1

1 (3, 3) 0 〈6〉+ 2A1 〈2〉+ 2〈−6〉+ A2 + A1

1 (3, 3) 1 〈6〉+ A1 + 〈−6〉〈2〉+ A2 + 2A1 + 〈−6〉1 (3, 3) 2 〈6〉+ 2〈−6〉 U+ 〈−6〉+ 3A1

1 (5, 3) 0 〈6〉+ D4 〈6〉+ A2(2) + 〈−6〉1 (5, 3) 1 〈6〉+ A2 + 2A1 〈2〉+ A1 + 2〈−6〉1 (5, 3) 3 〈6〉+ A2 + 2〈−6〉〈2〉+ 3A1

1 (5, 3) 2 〈6〉+ A2 + A1 + 〈−6〉〈2〉+ 2A1 + 〈−6〉1 (4, 4) 0 〈6〉+ 3A1 〈2〉+ A1 + 3〈−6〉1 (4, 4) 1 〈6〉+ 〈−6〉+ 2A1 〈2〉+ 2A1 + 2〈−6〉1 (4, 4) 2 〈6〉+ A1 + 2〈−6〉〈2〉+ 3A1 + 〈−6〉


1234567891011121314151617181920212223242526272829303132333435

Table 7A. Ascending T-pairs T± in the case of p = 0

δ2 (r, r2) q T+ T−0 (2, 0) 0 U U(3) + 2A2 + A1

0 (4, 0) 1 U+ A2 U(3) + A2 + A1

0 (6, 0) 2 U+ 2A2 U(3) + A1

0 (8, 0) 3 U+ 3A2 〈6〉1 (1, 1) 0 〈2〉 U(3) + 2A2 + 2A1

1 (3, 1) 0 U+ A1 〈6〉+ 〈−6〉+ 2A2

1 (3, 1) 1 U+ 〈−6〉 U(3) + A2 + 2A1

1 (5, 1) 1 U+ A2 + A1 〈6〉+ 〈−6〉+ A2

1 (5, 1) 2 〈2〉+ 2A2 U(3) + 2A1

1 (7, 1) 2 U+ 2A2 + A1 〈6〉+ 〈−6〉1 (7, 1) 3 〈2〉+ 3A2 〈6〉+ A1

0 (2, 2) 0 U(2) 〈6〉+ 〈−6〉+ 2A2 + A1

1 (2, 2) 0 〈2〉+ A1 〈6〉+ 〈−6〉+ 2A2 + A1

1 (2, 2) 1 〈2〉+ 〈−6〉〈6〉+ 2A2 + 2A1

0 (4, 2) 3 U(3) + A2(2) U+ A2(2) + A1

1 (4, 2) 0 U+ 2A1 〈6〉+ 2〈−6〉+ A2

1 (4, 2) 1 〈2〉+ A2 + A1 〈6〉+ 〈−6〉+ A2 + A1

1 (4, 2) 2 U+ 2〈−6〉 U(3) + 3A1

0 (4, 2) 1 U(2) + A2 〈6〉+ 〈−6〉+ A2 + A1

0 (6, 2) 2 U(2) + 2A2 〈6〉+ 〈−6〉+ A1

1 (6, 2) 2 U+ A2 + 〈−6〉+ A1 〈6〉+ 〈−6〉+ A1

1 (6, 2) 3 〈2〉+ 2A2 + 〈−6〉〈6〉+ 2A1

1 (6, 2) 1 U+ A2 + 2A1 〈6〉+ 2〈−6〉1 (3, 3) 0 〈2〉+ 2A1 〈6〉+ 2〈−6〉+ A2 + A1

1 (3, 3) 1 〈2〉+ A1 + 〈−6〉〈6〉+ 〈−6〉+ A2 + 2A1

1 (3, 3) 2 〈2〉+ 2〈−6〉 U(3) + 4A1

1 (5, 3) 3 U+ 3〈−6〉〈6〉+ 3A1

1 (5, 3) 2 〈2〉+ A2 + A1 + 〈−6〉〈6〉+ 〈−6〉+ 2A1

1 (5, 3) 1 〈2〉+ A2 + 2A1 〈6〉+ 2〈−6〉+ A1

1 (5, 3) 0 U+ 3A1 〈6〉+ 3〈−6〉1 (4, 4) 0 〈2〉+ 3A1 〈6〉+ 3〈−6〉+ A1

1 (4, 4) 1 〈2〉+ 2A1 + 〈−6〉〈6〉+ 2〈−6〉+ A1

1 (4, 4) 2 〈2〉+ A1 + 2〈−6〉〈6〉+ 〈−6〉+ 3A1

1 (4, 4) 3 〈2〉+ 3〈−6〉〈6〉+ 4A1

0 (4, 4) 3 U(6) + A2(2) 〈6〉+ 4A1

7.3.11. Theorem. Tables 7A-B give a complete list of the ascending T-pairs (T+, T−).

7.4. Property of (− 12)-transitivity for the eigenlattices T−. Here we establish a certain

property of T-halves T− to be used in the next section. We say that a lattice M is (− 12 )-

transitive if the automorphism group Aut(M) induces a transitive action on the non-characteristicelements x ∈ discr2(M) ⊂ discrM with qM (x) = − 1

2 . Note that for the characteristic element,v ∈ discr2(M) the value qM (v) is determined by the Brown invariant of discr2M . For instance, ifdiscr2M = p〈12 〉+q〈− 1

2 〉, then qM (v) = p−q2 mod 2Z, so, qM (v) = − 1

2 if and only if Br2(M) = 3mod 4.

7.4.1. Proposition. All the eigenlattices T− in the Tables 5 and 6A-B are (− 12 )-transitive.

Proof. First, note that the automorphisms of discr2 T− act transitively on the non-characteristic


(− 12 )-elements.

7.4.2. Lemma. For any elementary enhanced 2-group G the isometry group Aut(G) acts tran-sitively on the non-characteristic elements x ∈ G having q(x) = − 1

2 .

Proof. If V contains v ∈ G such that q(v) = − 12 , then q is odd and G = p〈12 〉 + q〈− 1

2 〉, wherep− q = Br(q). The orthogonal complement, Gv, of v is odd if and only if v is not characteristic,and in the latter case, Gv = 〈12 〉 + (q − 1)〈− 1

2 〉. Thus, for any other non-characteristic element

w ∈ G with q(w) = − 12 , we obtain an isomorphism Gv → Gw which is extended to G so that v

is sent to w.

It follows that 2-epistability implies (− 12 )-transitivity. So, we should analyze only the lattices

T− for which Nikulin’s stability criterion fails. Among these cases there are trivial ones, withdiscr2 T− containing only one or no non-characteristic (− 1

2 )-elements. This happens if r2(T−) 6 1,

or if discr2(T−) is isomorphic to 2〈12 〉, 〈12 〉+ 〈− 12 〉, 3〈12 〉, or 2〈12 〉+ 〈− 1

2 〉.So, it is sufficient to consider only those lattices T− non satisfying Nikulin’s criterion for which

discr2(T−) is 2〈− 12 〉, 〈12 〉 + 2〈− 1

2 〉, 3〈− 12 〉, or r2(T−) > 4. Analyzing the lattices T− in Tables 5

and 6A-B, in Section 7.2, we find that there remain only the following cases to consider:

(1) 〈6〉+ 3〈−6〉,(2) 〈2〉+ 3〈−6〉,(3) 〈6〉+ k〈−6〉+ A1, 0 6 k 6 3.

If T− = 〈6〉 + 3〈−6〉, then T−(13 ) is epistable by Nikulin’s criterion. In the cases (2) and (3)with k > 0, Proposition 3.12.2 is applicable.

In the remaining case, T− = 〈6〉+A1, the discriminant component discr2 T− = 2〈− 12 〉 has only

one non-identity automorphism. This automorphism interchanges the summands of discr2 T−,and therefore it is induced by the reflection ρv in T− with respect to v = (1, 1) ∈ T−. Thus, T−is 2-epistable and, in particular, (− 1

2 )-transitive.

8. Back to Zariski curves

8.1. Classification of geometric involutions.

8.1.1. Theorem. Geometric involutions c, c′ ∈ C(L,∆, h) have the same homological type (i.e.,[c] = [c′] ∈ C[L,∆, h]) if and only if the corresponding T-pairs (T+(c), T−(c)) and (T+(c

′), T−(c′))

are isomorphic (i.e., T±(c) ∼= T±(c′)).

Proof. The “only if” part is trivial. Assume that there are isomorphisms T±(c) ∼= T±(c′). With-

out loss of generality we may assume that r2(T+) < r2(T−), i.e., that the involutions c and c′ are

ascending. Using Proposition 3.13.2 we can observe that the restrictions c|T and c′|T to the sub-lattice T ⊂ L are conjugate via some automorphism fT : T → T . Namely, Corollary 7.3.2 showsthat either T− or T+ is epistable, and in the first case, the condition (1) of Proposition 3.13.2 issatisfied. In the case T+ is epistable, the condition (2) of Proposition 3.13.2 includes, in addition,a certain transitivity assumption; this assumption is satisfied according to Lemma 4.9.6 (imply-ing that transitivity of the action of Aut(T−) on the subgroups of discr2(T−) anti-isomorphic todiscr2(T+) is equivalent to (− 1

2 )-transitivity of this group action on T−), and Proposition 7.4.1(which establishes the latter transitivity).

Next, we extend fT to an automorphism fT ′ : T ′ → T ′ by letting h 7→ h (see Section 3.10).Since c(h) = c′(h) = −h, the restrictions c|T ′ and c′|T ′ are conjugate via fT ′ .

Finally, using Proposition 5.4.5 and Corollary 5.3.4 we conclude that fT ′ can be extended tof ∈ Aut(L,∆, h) that conjugates c with c′.


8.2. Reversion roots. Given an ascending T-pair (T1, T2), we say that an element v ∈ T2, isa reversion root for (T1, T2) if, first, it is an even (−2)-element (see the definitions in Section3.10), and, second, [ v2 ] ∈ discr2 T2 is a characteristic element in the case δ2(T1) = 0, and non-characteristic in the case δ2(T1) = 1. Due to Lemma 3.10.3) the latter condition implies thefollowing property of the orthogonal complement T v

2 ⊂ T2 of v.

8.2.1. Corollary. δ2(T1) = δ2(Tv2 ).

8.3. Reversion partners of T-pairs. Given an ascending T-pair (T1, T2) and a reversion rootv ∈ T2, we can write T2 = Zv+T v

2 . Interchanging T1 and Tv2 we obtain another pair (T v

2 ,Zv+T1),which will be called the reversion partner of (T1, T2) with respect to v. If a reversion root doesexist, we say that (T1, T2) is a reversible pair.

8.3.1. Lemma. The reversion partner (T ′1, T

′2) = (T v

2 , Tv) is also an ascending T-pair. More-over,

(1) r(T ′i ) = 8− r(Ti), i = 1, 2;

(2) r2(T′i ) = r2(Ti), and δ2(T

′i ) = δ2(Ti);

(3) discr3(T′1) = discr3(T2), discr3(T

′2) = discr3(T1), and, in particular, if discr3(Ti) = p〈23 〉+

q〈− 23 〉 then discr3(T

′i ) = (1− p)〈23 〉+ (3− q)〈− 2

3 〉.Furthermore, v ∈ T ′

2 is a reversion root of (T ′1, T

′2) and the reversion partner of (T ′

1, T′2) with

respect to v is (T1, T2).

Proof. From the construction of (T ′1, T

′2) it follows that r(T

′1) = r(T2)− 1 = (9− r(T1))− 1, and

r(T ′2)−1 = r(T1) = 9−r(T2), which gives (1). Next, r2(T

′1) = r2(T2)−1 = r2(T1), and Corollary

8.2.1 yields (2). Lemma 3.10.1 implies that discr3(v⊥) = discr3(T1), which implies (3). The last

claim is obvious from the construction.

8.3.2. Proposition. For any ascending pair (T1, T2), its reversion partner (T ′1, T

′2) if exists is

independent up to isomorphism of the choice of a reversion root v.

Proof. By Lemma 8.3.1 the invariants r, r2, δ2, p, and q of T′i are determined by those of Ti. These

invariants determine an ascending T-pair uniquely up to isomorphism, see Corollary 7.3.10.

8.3.3. Theorem. A reversion partner exists for all the ascending T-pairs except six pairs (T1, T2)listed in Table 8A. The partnership of the remaining 62 ascending T-pairs is shown in Tables8B-C, where partners are placed in the same rows.

123456

Table 8A. Irreversible T-pairs

δ2 (r, r2) (p, q) (r′, r′2) (p′, q′) T1 T2

—— —— —— —— —— ——— ———0 (8, 0) (0, 3) (1, 1) (1, 0) U+ 3A2 〈6〉1 (7, 1) (0, 2) (2, 2) (1, 1) U+ 2A2 + A1 〈6〉 + 〈−6〉0 (6, 2) (1, 1) (3, 3) (0, 2) U(3) + D4 〈6〉 + A2(2)1 (6, 2) (0, 1) (3, 3) (1, 2) U+ A2 + 2A1 〈6〉 + 2〈−6〉1 (5, 3) (0, 0) (4, 4) (1, 3) U+ 3A1 〈6〉 + 3〈−6〉1 (5, 3) (1, 0) (4, 4) (0, 3) 〈6〉+ D4 〈2〉 + 3〈−6〉

Proof. For the reversion partners the invariants r, r2, δ, p, and q should match as is indicated inLemma 8.3.1. For the six exceptional T-pairs in Table 8A there is no candidates to be a partnerwith the matching invariants. For the other T-pairs such a candidate is unique, since theseinvariants uniquely determine an ascending T-pair by Corollary 7.3.10. It is straightforward tocheck that the T-pairs placed in the same rows in Tables 8B-C have matching above invariants.Now, it is sufficient to check existence of a reversion root v ∈ T2 for every T-pair (T1, T2) in


123

456789

101112

1314151617181920

212223242526

27282930

31

Table 8B. The case of o = − (p = 0)

(r, r2) q δ2 T1 T2

(2, 0) 0 0 U U(3) + 2A2 + A1

(4, 0) 1 0 U+ A2 U(3) + A2 + A1

(6, 0) 2 0 U+ 2A2 U(3) + A1

(1, 1) 0 1 〈2〉 U(3) + 2A2 + 2A1

(3, 1) 0 1 U+ A1 〈6〉 + 〈−6〉 + 2A2

(3, 1) 1 1 U+ 〈−6〉 U(3) + A2 + 2A1

(5, 1) 1 1 U+ A2 + A1 〈6〉 + 〈−6〉 + A2

(5, 1) 2 1 〈2〉 + 2A2 U(3) + 2A1

(7, 1) 3 1 〈2〉 + 3A2 〈6〉 + A1

(2, 2) 0 0 U(2) 〈6〉+2A2+〈−6〉+A1

(2, 2) 0 1 〈2〉+ A1 〈6〉+2A2+〈−6〉+A1

(2, 2) 1 1 〈2〉 + 〈−6〉〈6〉+ 2A2 + 2A1

(4, 2) 2 1 U+ 2A1 〈6〉 + 2〈−6〉 + A2

(4, 2) 1 0 U(2) + A2 〈6〉+〈−6〉+A2+A1

(4, 2) 1 1 〈2〉 + A2 + A1 〈6〉+〈−6〉+A2+A1

(4, 2) 2 1 U+ 2〈−6〉 U(3) + 3A1

(4, 2) 3 0 U(3) + A2(2) U+ A2(2) + A1

(6, 2) 2 0 U(2) + 2A2 〈6〉 + 〈−6〉 + A1

(6, 2) 2 1 U+A2+〈−6〉+A1 〈6〉 + 〈−6〉 + A1

(6, 2) 3 1 〈2〉 + 2A2 + 〈−6〉〈6〉 + 2A1

(3, 3) 2 1 〈2〉 + 2A1 〈6〉+2〈−6〉+A2+A1

(3, 3) 1 1 〈2〉 + 〈−6〉 + A1 〈6〉+〈−6〉+A2+2A1

(3, 3) 2 1 〈2〉 + 2〈−6〉 U(3) + 4A1

(5, 3) 1 1 〈2〉+ A2 + 2A1 〈6〉 + 2〈−6〉 + A1

(5, 3) 2 1 〈2〉+A2+〈−6〉+A1 〈6〉 + 〈−6〉 + 2A1

(5, 3) 3 1 U+ 3〈−6〉〈6〉 + 3A1

(4, 4) 0 1 〈2〉 + 3A1 〈6〉 + 3〈−6〉 + A1

(4, 4) 1 1 〈2〉 + 〈−6〉+ 2A1 〈6〉 + 2〈−6〉 + 2A1

(4, 4) 2 1 〈2〉 + 2〈−6〉 + A1 〈6〉 + 〈−6〉 + 3A1

(4, 4) 3 1 〈2〉 + 3〈−6〉〈6〉 + 4A1

(4, 4) 3 0 U(6) + A2(2) 〈6〉 + 4A1

Table 8C. The case of o = + (p = 1)

(r, r2) T1 T2

(6, 0) U(3) + 2A2 U+ A1

(4, 0) U(3) + A2 U+ A2 + A1

(2, 0) U(3) U+ 3A2

(7, 1) 〈6〉 + 3A2 〈2〉 + A1

(5, 1) U(3) + A2 + 〈−6〉 U+ 2A1

(5, 1) U(3) + A2 + A1 U+ 〈−6〉+ A1

(3, 1) U(3) + 〈−6〉 U+ A2 + 2A1

(3, 1) 〈6〉 + A2 〈2〉 + 2A2 + A1

(1, 1) 〈6〉 U+2A2+A1+〈−6〉

(6, 2) U(6) + 2A2 〈2〉 + 2A1

(6, 2) U(3)+A2+A1+〈−6〉〈2〉 + 2A1

(6, 2) 〈6〉 + 2A2 + A1 〈2〉 + A1 + 〈−6〉(4, 2) U(3) + 2〈−6〉 U+ 3A1

(4, 2) U(6) + A2 〈2〉 + A2 + 2A1

(4, 2) 〈6〉 + A2 + 〈−6〉〈2〉 + A2 + 2A1

(4, 2) 〈6〉 + A2 + A1 〈2〉+〈−6〉+A2+A1

(4, 2) U+ A2(2) U(3) + A2(2) + A1

(2, 2) U(6) U+A2+2A1+〈−6〉(2, 2) 〈6〉 + 〈−6〉 U+A2+2A1+〈−6〉(2, 2) 〈6〉 + A1 〈2〉+2A2+A1+〈−6〉

(5, 3) 〈6〉+ A2 + 2〈−6〉〈2〉 + 3A1

(5, 3) 〈6〉+A2+A1+〈−6〉〈2〉+ 2A1 + 〈−6〉(5, 3) 〈6〉 + A2 + 2A1 〈2〉+ A1 + 2〈−6〉(3, 3) 〈6〉 + 2〈−6〉 U+ 〈−6〉 + 3A1

(3, 3) 〈6〉 + A1 + 〈−6〉〈2〉 + A2 + 2A1 + 〈−6〉(3, 3) 〈6〉 + 2A1 〈2〉 + 2〈−6〉 + A2 + A1

(4, 4) 〈6〉 + 3〈−6〉〈2〉 + 4A1

(4, 4) 〈6〉+ A1 + 2〈−6〉〈2〉+ 3A1 + 〈−6〉(4, 4) 〈6〉+ 〈−6〉 + 2A1 〈2〉 + 2A1 + 2〈−6〉(4, 4) 〈6〉 + 3A1 〈2〉+ A1 + 3〈−6〉

(4, 4) U(2) + A2(2) 〈2〉+ A1 + 3〈−6〉

Table 8B (this implies existence of a reversion root in the partner T-pair in Table 8C). This isalso straightforward and easy.

8.3.4. Proposition. If ascending T-pairs (T+(c), T−(c)) and (T+(c′), T−(c

′)) associated withZariski curves, A and A′, are reversion partners, then A and A′ are reversion partners themselves.

Proof. Lemma 4.9.6, which reduces the transitivity of the action of Aut(T−) on the subgroups ofdiscr2(T−) anti-isomorphic to discr2(T+) to the transitivity on the elements of discr2 T− whosesquare is − 1

2 , and Proposition 7.4.1, which establishes the latter transitivity, imply that h isglued with a reversion root v. Thus, h and v generate an U-summand in T ′

−. Therefore, c′ = c∨

in the sense of Section 6.4. Applying Proposition 6.4.1 we find an example of reversion partners,B and B′, that have the same homological types as A and A′, respectively. By Theorem 8.1.1and Theorem 6.3.1, B is deformation equivalent to A and B′ to A′.

8.4. Classification of real Zariski sextics. Given a real Zariski sextic A, we associate to

it a geometric involution c ∈ C(L,∆, h) induced on a K3-lattice L = H2(Y ) with a conical

(∆, h)-decoration by the ascending real structure, conjY: Y → Y (see 4.1, 4.8). The homology

type [c] ∈ C[L,∆, h] depends only on the deformation class [A] of A, and the mapping [A] 7→ [c]


gives by Theorem 6.3.1 a one-to-one correspondence between the set of deformation classes ofreal Zariski sextics and the set C<[L,∆, h] = [c] ∈ C[L,∆, h] | c is ascending. Then we obtaina mapping that sends [c] to the isomorphism type of (T+(c), T−(c)).

8.4.1. Theorem. Associating with a real Zariski sextic A the eigenlattices (T+(c), T−(c)) of c ∈C<(L,∆, h) defined by the ascending real structure in Y , we obtain a one-to-one correspondencebetween the deformation classes of real Zariski sextics and the ascending T-pairs listed in theTables 8A, 8B and 8C.

Proof. Theorem 8.1.1 shows injectivity of the map from C<[L,∆, h] to the set of isomorphismtypes of ascending T-pairs (T+(c), T−(c)), and Theorem 5.5.6 shows its surjectivity. Theorem7.3.11 enumerates the ascending T-pairs (they are listed first in Tables 7A-B, and then presentedin a different order in Tables 8A-B to fit to our final description of the IDs of real Zariski sexticsin Theorem 2.9.1).

8.5. The IDs of real Zariski sextics. Our aim now is to obtain from Theorem 8.4.1 anenumeration of the deformation classes of real Zariski sextics A in terms of their IDs. Thefollowing lemma describes how the characteristics ℓ(A), χ(A−), νr(A), o(A), and the type of Aare expressed in terms of the invariants r, r2, δ2, p and q of T+(c)

8.5.1. Lemma. The ID of a real Zariski sextic A determines the values r, r2, δ2, p and q of

the eigenlattices T± ⊂ L = H2(Y ) associated with the ascending real structure in Y . Conversely,the values r, r2, δ2, p and q determine ℓ(A), χ(An), νr, the type of A (I or II), and the signo(A). Namely, if A(R) 6= ∅, then the following relations hold.

(1) r2(T+) = r2(T−)− 1 = 5− ℓ(A), r(T+) = 9− r(T−) = 4 + χ(An),in particular, in the case of code α+1〈β〉, we have r2(T+) = 4−(α+β), r(T+) = 4+(β−α),or equivalently α = 4− r+r2

2 and β = r−r22 .

(2) The type of A determines δ2(T+) and is determined by it, namely, δ2(T+) = 0 if A hastype I, and δ2(T+) = 1 if type II.

(3) The sign o(A) and the number of real cusps, 2νr(A), determine 0 6 p 6 1 and 0 6 q 6 3,and conversely, p and q determine o(A) and 2νr as follows.

(p, q) =

(0, 3− νr(A)), if o(A) = −,(1, νr(A)), if o(A) = +,

(o(A), νr) =

(−, 3− q), if p = 0,

(+, q), if p = 1.

If A(R) = ∅, then r(T+) = r2(T+) = 4, δ2(T+) = 0, and (p, q) is (1, 0) if o(A) = − and (0, 3)if o(A) = +.

Proof. In the case A(R) 6= ∅, the ascending involution is the Mobius one, see Lemma 4.8.2. So,the relations for r2(T±) in (1) follow from Lemma 4.2.1 and Corollary 4.2.4. The relation betweenr(T+) and χ(An) follows from Lemmas 2.8.1 and 4.1.1. Item (2) follows from Proposition 4.3.5,and (3) from Corollary 4.9.4.

In the case of A(R) = ∅, the values of r, r2 and δ2 for T+ are found in Proposition 4.10.1, andthe relation for (p, q) are obtained from (3) by alternation of the sign o(A), because the ascendingreal structure is non-Mobius in the case of A(R) = ∅.

8.5.2. Lemma. Assume that A is a real Zariski sextic and the covering desingularized K3-

surface Y is endowed with the ascending real structure. Then the following conditions are equiv-alent:

(1) A(R) = ∅;

(2) Y (R) = ∅;

(3) the eigenlattice T+(c) of the involution c induced in T ⊂ H2(Y ) by the real structure iseither U(2) + A2(2), or U(6) + A2(2);


(4) the ascending T-pair (T+(c), T−(c)) is either the one in the last row of Table 8B, or theone in the last row of Table 8C.

Proof. For A(R) 6= ∅, we have Y (R) 6= ∅ by definition, and for A(R) = ∅ the ascending real

structure in Y is the non-Mobius one (see Lemma 4.8.2), with Y (R) = ∅, which shows equivalenceof (1) and (2).

Proposition 4.10.1 says that (1) implies r(T+) = r2(T+) = 4, and δ2(T+) = 0. Let discr3 T+ =p〈23 〉 + q〈− 2

3 〉, then νr = 0 implies that (p, q) is either (1, 0), or (0, 3) depending on o(A) (seeLemma 8.5.1). These characteristics detect precisely two ascending T-pairs from Tables 7A-B,which are characterized by T+(c) in (3). Existence of at least two deformation classes of “empty”Zariski sextics A (with o(A) = ±) shows that such T-pairs really correspond to the case of

Y (R) = ∅. The aforementioned T-pairs are placed in the last rows of Tables 8B and 8C.

Remark. Recall that in the case A(R) = ∅, the central projection of the cubic surface onto P 2(R)is one-to-one if o(A) = − and three-to-one if o(A) = +. ⊡

8.5.3. Theorem. The ID of a real Zariski sextic whose equisingular deformation class is deter-mined by the associated eigenlattices (T+, T−) in one of the Tables 8A, 8B, and 8C, is listed inthe same row of the Table 1A, 1B, and 1C respectively.

Proof. It was already shown in 8.5.2 that for a real Zariski sextic A with A(R) = ∅ the associatedpair (T+, T−) is like in the last rows of Tables 8B and 8C (depending on o(A) ∈ +,−).So, in what follows we assume that A(R) 6= ∅ and exclude the last row of Tables 8B-C fromconsideration.

G.Mikhalkin [M] found 49 complete codes of real Zariski sextics A indicating the topologicaltype of the cubics X(R) whose apparent contours they do represent. As follows from Lemma2.7.1, χ(X(R)) together with the complete code of A detect o(A). Thus, using Lemma 8.5.1,one can determine the invariants r, r2, p and q for the corresponding lattices T±, while δ2(T+)remains unknown, since the type of A was not determined by Mikhalkin. Reviewing Tables 7A-Bwe can observe that the above invariants r, r2, p and q distinguish all the rows there except sixpairs of rows. In each of these pairs, lattices T+ are distinguished by the value of δ2.

The ascending T-pairs (T+, T−) whose invariants r, r2, p and q do not match the real Zariskisextics found by Mikhalkin, appear in the rows 4 and 6 of Table 8A, rows 2, 3, 7, 8 of Table 8B,and rows 1, 2, 5, 6, 7, 13 and 20 of Table 8C.

If we know the ID of a real Zariski sextic A associated with (T+, T−), then we may determinethe ID of the one associated with the partner of (T+, T−) using Proposition 8.3.4, if there is anon-empty oval in A(R), see Corollary 2.6.2. Tables 8B and 8C are arranged to place the partnerpairs (T+, T−) in the same rows. The sextics A represented by (T+, T−) in rows 1, 5, 6, 13, 20 ofTable 8B and rows 3, 8 of Table 8C turn out to have a non-empty oval, so it remains to determinethe IDs of A represented by (T+, T−) from Table 8A, rows 4 and 6 (since they have no partners),and from Tables 8B and 8C, rows 2 and 7 (since Mikhalkin’s examples are missing for the bothpartners).

In the remaining part of the proof we may suppose that the simple code A looks like α⊔ 1〈β〉,since the case of null-code was considered in the beginning and the two cases of 3-nest codes(with different values of o(A)) are represented in the row 17 of Tables 8B and 8C, which requiresno further analysis (it follows from Lemmas 2.4.2 and 2.5.1 that these are the only IDs with the3-nest code).

Lemma 8.5.1 shows how to reconstruct from (T+, T−) the type of A, νr(A), and o(A). So, itremains only to determine the distribution of the real cusps on the ovals of A(R) to reconstructits ID.


8.5.4. Lemma. The real Zariski sextics A characterized by the eigenlattices (T+, T−) in rows4 and 6 of Table 8A and rows 2 and 7 in each of Tables 8B–8C have IDs indicated in thecorresponding rows of Tables 1A and 1B–1C respectively.

Proof. In all the cases indicated A has type II, since δ2(T+) = 1. In the case of row 6 of Table8A, Lemma 8.5.1 says that A has no real cusps and thus, the complete code coincides with thesimple one. This Lemma says also that this code is 1〈1〉, and determines the sign o(A) = +(since p = 1), giving the ID indicated in row 6 of Table 1A.

For row 4 of Table 8A, we get νr = 1, the simple code 1〈2〉, and o(A) = −. Lemma 2.3.1implies that the both real cusps should lie on the same oval, by Lemma 2.3.2, they must beoutward cusps if lying on the ambient oval and inward cusps in lying on an internal one. But thelatter case is forbidden by Lemma 2.3.5, so the complete code of A must be 11〈2〉.

In the case of row 2 of Table 8B, we similarly find νr = 2, the sign o(A) = −, and the simplecode 2⊔1〈2〉. Applying Lemmas 2.3.2 and 2.3.5, we conclude that the two pairs of outward cuspsmust be distributed among the external ovals and the ambient oval. Lemmas 2.3.6 and 2.7.3exclude a possibility that one of the external ovals is smooth, thus both of them have a pair ofcusps and the complete code is 21 ⊔ 1〈2〉.

In the case of row 7 of Table 8B, we find νr = 2, o(A) = −, and the simple code 1 ⊔ 1〈2〉.As above, we conclude that there should be two pairs of outward cusps distributed among theambient and the external ovals. Lemma 2.7.3 exclude possibility that all the cusps lie on theambient oval, while Lemma 2.3.5 exclude possibility that all the cusps lie on the exterior one.Thus, the complete code must be 11 ⊔ 11〈2〉.

The rows 2 and 7 of Table 8C represent the reverse partners for the same rows of Table 8B,and thus, the corresponding curves A(R) must be in reverse positions by Proposition 8.3.4. Sincenot all the ovals are empty, the complete code of the reverse partners can be found by the rulein Corollary 2.6.2, namely, reversion of 21 ⊔ 1〈1〉 gives 1 ⊔ 1〈21〉 and reversion of 11 ⊔ 11〈2〉 gives2 ⊔ 1−1〈11〉, like indicated in Tables 1C.

Finally, it remains to analyze the six pairs of rows which differ only by the value of δ2(T+),namely, rows 10 with 11, rows 14 with 15, and rows 18 with 19 in Table 8B, and the same pairsof rows in Table 8C.

8.5.5. Lemma. Each of the three pairs of rows, 10 with 11, 14 with 15, 18 with 19 in Table 8B,represent a pair of deformation classes of real Zariski sextics A which have the same completecodes. The same is true for the same pairs of rows in Table 8C. The corresponding completecodes are like indicated in the corresponding rows of Tables 1B and 1C.

Proof. Using Lemma 8.5.1 like in the proof of Lemma 8.5.4, we conclude that the complete codesof A for the indicated pairs of rows may differ only by the distribution of cusps on the ovals,since the values of r, r2, p and q for T+ in each of the pairs of rows is the same.

Rows 10 and 11 of Table 8B give both the simple code 3, with νr = 3 and o(A) = −, whichimplies that there are three pairs of outward cusps. None of these three ovals can be smooth byCorollary 2.7.4, so, both rows should give complete code 31.

Rows 14 and 15 give both the simple code 1 ⊔ 1〈1〉, with νr = 2, and o(A) = −. The signo(A) together with Lemmas 2.3.2 and 2.3.5 imply that the cusps are outward (cf. the proof ofLemma 8.5.4). An external oval can be neither smooth nor 4-cuspidal by Lemmas 2.7.3 and 2.3.5respectively. So, the complete code for the both rows is 11 ⊔ 11〈1〉.

Rows 18 and 19 give the simple code 1〈2〉 with νr = 1 and o(A) = −. This implies that thereis one pair of cusps which must lie on the ambient oval, since Lemmas 2.3.2 and 2.3.5 excludeother possibilities.

The same rows in Table 8C represent the reversion partners for the ones considered above,and thus, their complete code is obtained by reversion, according to the rules in Corollary 2.6.2.


As the result, we can deduce that in each of these six pairs, the cusps must be distributed onthe ovals in the same way, and thus, the corresponding real Zariski sextics A must have the IDslike indicated in the 1B and 1C.

8.6. Proof of Theorem 2.9.1. Theorem 8.4.1 together with the correspondence establishedin Theorem 8.5.3 imply Theorem 2.9.1.

9. Concluding Remarks

9.1. Purely real statements. Although the ID of a real Zariski sextic refers to its complexpoint set, it is sufficient to look at the Tables 1A-C to conclude that in the majority of cases thedeformation classes are determined only by the complete codes, which are purely real invariants.There are however a few exceptions. The first group of exceptions is given by real Zariski sexticswith the complete codes ∅, 1, 1〈1〉, and 1〈1〈1〉〉. In each of these cases there are 2 deformationclasses. But they can be distinguished by enhancing the code with the invariant o(A) = ±, or inother words, marking the domain where the projection of the cubic surface is three-to-one andthus expressing the classification in terms of “purely real” data.

The other group of exceptions contains 6 complete codes, each one again representing 2 de-formation classes, but this time the classes in each pair differ by the types, I or II, of the sextics.The only remedy we can suggest to distinguish the types in purely real terms is to use real linespassing through a pair of ovals of the sextic. In one case such lines separate the cusps on thethird oval as it is shown on Figure 4, and in the other, they do not.

Figure 4. The six pairs of codes that differ only by their type (the bottom pairs arethe partners of the top pairs)

type I type I type Itype II type II type II

Figure 5. Construction of the curves on Figure 4

It would be interesting to find a conceptual explanation to this observation. Our current proofthat the types are as indicated on Figure 4, is based on the construction of these curves shownon Figure 5 (the construction in question consists in a small perturbation, p2 + εq3, where p is areducible cubic curve and q is a conic). Since sextics-partners have the same type, it is sufficientto consider one representative from each pair of partners.


Recall that according to our definitions, a real Zariski sextic is the apparent contour of a genericprojection of a nonsingular real cubic surface, where the latter is supposed to have neither realnor complex singular points and a generic apparent contour is supposed to have no singular pointsother than ordinary (real or complex) cusps. Therefore, it is natural to ask how the deformationclassification of generic apparent contours may change if we allow for cubic surfaces to have non-real singular points and for Zariski sextics to have non-real singularities other than cusps. Theanswer is straightforward, it says that the deformation classification does not change. (However,to make the formulation and solution of the problem depend only on the real locus of Zariskisextics in question, it is better to exclude the case of sextics with empty real locus; in all theother cases, the real locus is sufficient for extending the real central projection correspondencestated in Proposition 2.2.1 to this new setting.).

9.2. Transversal pairs of conic and cubic. As is known, see Proposition 2.1.4, a Zariskisextic is uniquely defined by a pair of homogeneous polynomials of degree 2 and 3 defining aconic and a cubic intersecting transversely. Thus, it may be worth to ask about deformationclassification of such pairs of polynomials. Understanding this question literally, that is as aclassification of pairs p, q where p, q are a conic and a cubic (not necessarily nonsingular) inter-secting transversally each other, one easily gets the following answer. Over C, there is only onedeformation class. And over R, there are 4 classes; they are distinguished just by the number ofreal intersections points: 6, 4, 2, or none.

Curiously enough, the latter result being so far from the principal object of our investigationis however also related to classification of cubic surfaces. Namely, if we impose the assumptionof non-singularity on the conic, then aforesaid transversal pairs describe cubic surfaces with onenode. Over C the pairs of homogeneous polynomials of degree 2 and 3 defining a non-singularconic and a (possibly singular) cubic intersecting its transversely form a single deformation class,while over R we get 10 deformation classes, more than before, because of the well defined sign ofthe degree 2 polynomial on the nonorientable part of the complement of the set of its zeros in thereal projective plane (now, the sign can not be changed, because we have forbidden to the conicto become singular) and a possibility to have 0 intersection points in two ways, with empty andwith non empty conic. As a consequence, real cubic surfaces with one node form 5 deformationclasses (twice less, since reversing of the sign of the polynomial of the cubic surface leads to thechange of the sign of the polynomial defining the conic); the fact which was observed exactly inthis way by F. Klein, who used it as a step to his classification of real nonsingular cubic surfacesup to deformation.

A different, but somehow related and typical problem is the classification of pairs of transver-sally intersecting nonsingular curves. As was shown by G. Polotovsky [Pol], in the case of a realconic and a real cubic one obtains 25 deformation classes: each deformation class is determinedby the topology of the arrangement of real points in the real projective plane; there are 7 extremalclasses shown in Figure 6 and the other ones are obtained from them by two moves: erasing anoval (of the cubic or of the conic) containing no intersection points, and shifting a piece of curvecontaining a pair of consecutive (both on the conic and the cubic) intersection points, so thatthese two intersections disappear.

Figure 6. The extremal mutual positions of a cubic and a conic

The both classification problems for the pairs formed by a conic and a cubic are different from


the problem of classification of Zariski sextics by an evident, although a deep, reason. In the caseof Zariski curves there is a more subtle non-singularity condition: the sextic p2 + q3 should nothave singular points other than the intersection points of the conic q with the cubic p.

9.3. Ordering of IDs. To characterize the set of complete codes of Zariski sextics one can givea short list of the extremal ones, from which all the others can be obtained by certain simplifyingmoves. The extremal codes include seven M-codes, namely, 1〈4〉, α1 ⊔ 1〈β〉, and α ⊔ 1〈β1〉,1 6 α, β 6 3, α+ β = 4, the 3-nest code 1〈1〈1〉〉, and the code 1〈11 ⊔ 1〉. The simplifying movesare:

(1) cancelation of an empty smooth oval (for instance, 2 ⊔ 1〈21〉 gives 1 ⊔ 1〈21〉),(2) cancelation of an empty oval with a pair of outward cusps (for instance, 2 ⊔ 1〈21〉 gives

2 ⊔ 1〈11〉),(3) fusion of an empty oval that has a pair of outward cusps with a principal oval (the latter

was defined in Section 2.6) as is shown on Figure 7 (for instance, 21⊔1〈2〉 gives 11⊔11〈2〉,while 2 ⊔ 1〈21〉 gives 2 ⊔ 1−1〈11〉).

Figure 7. Fusion moves

All the complete codes of Zariski sextics can be obtained by such moves from the extremalones. However, one can obtain also a few extra ones, which are not the codes of Zariski sextics.Namely, 1⊔1−3, 2⊔1−2, and 3⊔1−1, are obtained by fusion-move applied to all the internal ovalsof α ⊔ 1〈(4 − α)1〉, α = 1, 2, 3. These are the only three exceptions: if we apply simplificationmoves to any of the extremal codes in any other way, then we necessarily obtain again a code ofa Zariski sextic.

9.4. Nonsingular partners. The partner duality that played an important role in our classifi-cation of real Zariski sextics (in matching the homological types against the IDs) exists as well inthe case of nonsingular real sextics. Implicitly, it appears already in Hilbert’s sixteenth problemstatement. Furthermore, it performed a dramatic and somehow decisive role in Gudkov’s clas-sification of real nonsingular sextics. It was the subject of his habilitation thesis, and when hehad shown the preliminary version to one of his ”thesis referees”, V.V. Morozov (Professor at theKazan University), the latter have objected the resulting classification exactly because of a smallirregularity with respect to the ”reversion symmetry” in it. It is by repairing this asymmetrythat Gudkov has come to his final result. Such a reversion symmetry revealed itself forcefullyagain in Rokhlin’s and Nikulin’s treatment of deformation classes of real nonsingular sextics. Upto the best of our knowledge, a conceptual explanation of this partner duality/reversion symme-try was never explicitly presented in the literature. In fact, such an explanation for nonsingularsextics is literarily the same as for Zariski sextics, both in the lattice-arithmetical and in geomet-ric terms. Namely, for each deformation class with one exception, the eigenlattice L− containsan U-summand and the lattice-arithmetical form of the partner duality consists in transferringthe U-summand to the opposite eigenlattice L+ and then exchanging of the eigenlattices. Ingeometrical terms it means that each partner in a partner pair can be deformed to a triple conic,near the triple conic the family looks as Q3 + tf2Q

2 + t2f4Q + f6 = 0 (cf., Introduction), andswitching of the sign of t (which corresponds to passing through the triple conic) replaces the


curves from one deformation class by the curves from its partner class. The only exceptionaldeformation class having no partner class is formed by real sextics of type I with the code 1〈4〉.9.5. Promiscuity. In the case of real sextics with arbitrary singularities, an analogue of thepartnership relation can be also defined, but it looks at first glance rather like a sort of promis-cuity, at least from one side. For instance, in the case of sextics with a node (cf., [It]), there is awell-defined partnership map (neither injective nor surjective) that transforms the deformationclasses of real sextics with an internal node to the ones with an external node as is illustrated onFigure 8.

Figure 8. The partnership map for real nodal sextics

If we shadows have offended,

Think but this, and all is mended,

That you have but slumber’d here

While these visions did appear.

And this weak and idle theme,

No more yielding but a dream,

Gentles, do not reprehend:

if you pardon, we will mend ...

“Midsummer Night Dream”, W. Shakespeare

References

[A] V. I. Arnold, On the teaching mathematics, Russian Math. Surveys 53 (1998), no. 1, 229–236.[BLLV] J. Barge, J. Lannes, F.Latour, P. Vogel, Λ-spheres, Annales scientifiques de l’Ecole Normale Superieure,

Ser. 4 7 (1974), no. 4, 463–505.

[BR] D.Burns, M.Rappaport, On the Torelli problem for kahlerian K3 surfaces, Annales scientifiques de l’EcoleNormale Superieure, Ser. 4 8 (1975), no. 2, 235–273.

[CF] C. Cilberto, F. Flamini, On the branch curve of a general projection of a surface to a plane, Trans.Amer. Math. Soc. 363 (2011), no. 7, 3457–3471.

[D1] A.Degtyarev, Oka’s conjecture on irreducible plane sextics, J. London Math. Soc. 78 (2008), no. 2,329–351.

[D2] A.Degtyarev, On deformations of singular plane sextics, J. Algeb. Geom. 17 (2008), 101–135.

[G] D. A. Gudkov, The topology of real projective algebraic varieties, Russian Math. Surveys 29 (1974),no. 4, 1-79.

[GM] L. Guilou, A. Marin, A la recherche de la topologie perdue, Birkhauser, 1986, pp. 97–118.

[It] I. Itenberg, Rigid isotopy classification of curves of degree 6 with one nondegenerate double point, Adv.in Sov. Math. 18 (1994), 193 - 208.

[J] D. G. James, Diagonalizable indefinite integral quadratic forms, Acta Arith. 50 (1988), 309 – 314.

[Kh1] V. Kharlamov, New congruences for the Euler characteristic of real algebraic varieties, Funkz. Anal. iPriloz. (1973), no. 2, 74–78.

[Kh2] V. Kharlamov, Topological types of real nonsingular surfaces of degree 4 in RP 3, Funkz. Anal. i Priloz.(1976), no. 4, 55-68.

[Ku1] Vik. Kulikov, On Chisini’s conjecture, Izv. Math. 63 (1999), no. 6, 1139-1170.

[Ku2] Vik. Kulikov, Surjectivity of the period map for a K3 surface, Usp. Mat. Nauk 32 (1977), no. 4, 257-258.[M] G. Mikhalkin, Visible contours of cubic surfaces in RP 3, Preprint MPI (1995), no. 95-34, 1-8.

[MM1] R. Miranda, D. Morrison, The number of embeddings of integral quadratic forms, I, Proc. Japan Acad.Ser. A Math. Sci. 61 (1985), no. 10, 317320.


[MM2] R. Miranda, D. Morrison, The number of embeddings of integral quadratic forms, I, Proc. Japan Acad.Ser. A Math. Sci. 62 (1986), no. 1, 29-32.

[Mo] Moishezon, Stable branch curves and braid monodromies, Lecture Notes in Math. 862 (1981), 107-192.[N1] V. V. Nikulin, Integer quadratic forms and some of their geometrical applications, Math. USSR – Izv.

43 (1979), 103–167.[N2] V. V. Nikulin, Discrete reflection groups in Lobachevsky spaces and algebraic surfaces, Proceedings of

the International Congress of Mathematicians (1986, Berkeley, Calif.), 1987, pp. 654–671.[N3] V. V. Nikulin, Involutions of integer quadratic forms and their applications to real algebraic geometry.

(Russian), Izv. Akad. Nauk SSSR Ser. Mat. 47 (1983), no. 1, 109–188.[O] M. Oka, Geometry of Reduced Sextics of Torus Type, Tokyo J. Math. 26 (2003), no. 2, 301–327.[OP] M. Oka, D. T. Pho, Classification of Sextics of Torus Type, Tokyo J. Math. 25 (2002), no. 2, 399–433.[Pho] D. T. Pho, Classification of singularities on torus curves of type (2,3), Kodai Math. J. 24 (2001), no. 2,

259–284.[Pol] G. M. Polotovsky, Full classification of M-decomposable real curves of order 6 in real projective plane.

(Russian), Dep. VINITI (1978), no. 1349-78, 1–103.[Sal] G. Salmon, A Treatise on the Analytic Geometry of Three Dimensions, Hodge, Smith, & Co., 1982.[SD] B. Saint-Donat, Projective models of K-3 surfaces, Am. J. Math. 96 (1974), 602–639.[Seg] B. Segre, Sulla caratterizzazione delle curve di diramazione dei piani multipli generali., Memorie Accad.

d’Italia 1 (1930), no. 4, 5–31.[U] T. Urabe, Dynkin graphs and combinations of singularities on plane sextic curves, Contemp. Math. 90

(1989), 295–316.[VdB] F. Van der Blij, An invariant of quadratic forms modulo 8, Indag/ Math. 21 (1959), 291-293.[W1] C.T.C.-Wall, Quadratic forms in finite groups and related topics, Topologty 2 (1964), 281–298..[W2] C.T.C.Wall, On the Orthogonal Groups of Unimodular Quadrat ic Forms, Math Annalen 147 (1962).[Z1] O. Zariski, On the Problem of Existence of Algebraic Functions of Two Variables Possessing a Given

Branch Curve., American Journal of Mathematics 51 (1929), no. 2, 305-328.[Z2] O. Zariski, Algebraic Surfaces, Ergebnisse der Mathematik und ihrer Grenzgebiete, Springer, 1935.

Middle East Technical University, Department of MathematicsAnkara 06800 Turkey

Universite de Strasbourg et IRMA (CNRS)7 rue Rene-Descartes 67084 Strasbourg Cedex, France

arxiv.org · arxiv:1306.2002v3 [math.ag] 9 feb 2015 apparent contours of nonsingular real cubic...

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